Surveying MNP MKL Given Segment NM Segment KM –Definition of a midpoint LMK PMN –Vertical Angles Theorem KLM NPM –ASA Congruence Segment LK Segment PN –Corresponding Parts of **Congruent** **Triangles** Lesson 5.3 Similar **Triangles** Ratio If a and b are two quantities measured in the same units, then the ratio of a to b is a/b.a/b. It can also/

obtuse angle in a **triangle**. **Congruent** **triangles** are **triangles** that are the same size and shape. **Congruent** **triangles** have the corresponding six parts (three angles, three sides) **congruent**. Definition: Two **triangles** are **congruent** if and only if their corresponding parts are **congruent**. Congruence of **triangles** is reflexive, symmetric, and transitive. Proving **Triangles** **Congruent** SSS: If the sides of one **triangle** are **congruent** to the sides of a second **triangle**, then the **triangles** are **congruent**. Given STU with/

). And no screaming! What did you learn today? What are the five ways (one for right **triangles**) to prove **triangles** are **congruent**? So what do we know about the parts of **congruent** **triangles**? **Congruent** Parts of **Congruent** **Triangles** are **Congruent** Hence, *Remember, you can only use CPCTC, AFTER you have proven two **triangles** to be **congruent**! CPCTC Song (sung to the tune of “YMCA” by the Village People) Author of lyrics/

B DD C EE AB FD BC DE AC FE These relationships help define the **congruent** **triangles**. Definition of **Congruent** **Triangles** If the _________________ of two **triangles** are **congruent**, then the two **triangles** are **congruent**. corresponding parts If two **triangles** are _________, then the corresponding parts of the two **triangles** are **congruent**. **congruent** ΔRST ΔXYZ. Find the value of n. T S R Z X Y 40° (2n + 10)° 50/

2 column proof: Given: óBAE óEDB, óABE óDEB Prove: óABE óDEB StatementsReasons So what do we know about the parts of **congruent** **triangles**? Corresponding Parts of **Congruent** **Triangles** are **Congruent** Hence, *Remember, you can only use CPCTC, AFTER you have proven two **triangles** to be **congruent**! Write a Proof Statement 1.FJ GH JFH GHF 2.HF FH 3. JFH GHF 4.FG JH Reasons 1.Given/

Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry 4-4 **Congruent** **Triangles** Use properties of **congruent** **triangles**. Prove **triangles** **congruent** by using the definition of congruence. Objectives Holt McDougal Geometry 4-4 **Congruent** **Triangles** corresponding angles corresponding sides **congruent** polygons Vocabulary Holt McDougal Geometry 4-4 **Congruent** **Triangles** Geometric figures are **congruent** if they are the same size and shape. Corresponding angles and corresponding sides are/

the same length. Holt McDougal Geometry 4-4 **Congruent** **Triangles** Use properties of **congruent** **triangles**. Prove **triangles** **congruent** by using the definition of congruence. Objectives Holt McDougal Geometry 4-4 **Congruent** **Triangles** Geometric figures are **congruent** if they are the same size and shape. / off the use Step 4: Get to the end goal, the PROVE Holt McDougal Geometry 4-4 **Congruent** **Triangles** Example 3: Proving **Triangles** **Congruent** Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of /

match up their vertices so that all pairs of CORRESPONDING ANGLES and all pairs of CORRESPONDING SIDES are **CONGRUENT**. **Congruent** Polygons **Triangle** ABC and **Triangle** PQR are **congruent** **Triangles** **Congruent** Polygons Use the figures to complete each statement. PQ ____ C ____ ABC ______/D A E Jim Smith JCHS Sections 4-2, 4-3, 4-5 When we talk about **congruent** **triangles**, we mean everything about them Is **congruent**. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal/

to 180 o. Acute **Triangles**, are **triangles** that have three acute angle. Examples: Obtuse **Triangles**, are **triangles** that have one obtuse angle. Examples: Right **Triangles**, are **triangles** that have one right angle. Examples: Scalene **Triangles**, are **triangles** that have no **congruent** sides or angles. Examples: Isosceles **Triangles**, are **triangles** with at least two **congruent** sides and two **congruent** angles. Examples: Equilateral **Triangles**, are **triangles** with three **congruent** sides and three **congruent** angles. Examples: More/

-> Side-Angle-Side (SAS) Postulate If two sides and the included angle of one **triangle** are **congruent** to two sides and the included angle of another **triangle**, then the two **triangles** are **congruent**. A B C D E F **Triangle** ABC **congruent** to **triangle** DEF Example using SAS Given: AB **congruent** BEBC **congruent** BD Prove: **Triangle** ABC **congruent** **triangle** DBE A B E C D Homework #20 Due Monday (Oct 22) Page 208/

AAS use corresponding parts to prove **triangles** **congruent**. CPCTC uses **congruent** **triangles** to prove corresponding parts **congruent**. Remember! Review Example 10: Using CPCTC A and B are on the edges of a ravine. What is AB? One angle pair is **congruent**, because they are vertical angles. Two pairs of sides are **congruent**, because their lengths are equal. Therefore the two **triangles** are **congruent** by SAS. By CPCTC, the third/

of supplementary angles. You Try: Find the Measure of the Unknown Angles 1 2 3 **Triangle** Congruence **Congruent** **triangles** are exactly the same size (same side lengths and angle measures) Corresponding sides are the **congruent** sides of **congruent** **triangles** Corresponding angles are the **congruent** angles of **congruent** **triangles** Examples of **Congruent** **Triangles** Note: We must name the **congruent** **triangles** correctly according to the corresponding angles! You Try: Write a Statement Indicating the Pairs/

CPCTC is an abbreviation for the phrase “Corresponding Parts of **Congruent** **Triangles** are **Congruent**.” It can be used as a justification in a proof after you have proven two **triangles** **congruent** because by definition, corresponding parts of **congruent** **triangles** are **congruent**. 4-4 Prove **Triangles** **Congruent** The Basic Idea: Given Information SSS SAS ASA AAS Prove **Triangles** **Congruent** Show CorrespondingParts **Congruent** CPCTC Using **Congruent** **Triangles**: CPCTC GEOMETRY LESSON 4-4 SSS, SAS, ASA, AAS, (and/

AND SAS CONGRUENCE POSTULATES If all six pairs of corresponding parts (sides and angles) are **congruent**, then the **triangles** are **congruent**. Sides are **congruent** Angles are **congruent** **Triangles** are **congruent** If and then ABC DEF 1. AB DE 4. A D 2. /postulate SSS postulate Not Enough Info SAS postulate SAS postulate Use the SSS Congruence Postulate to show that NMP DEF. **Congruent** **Triangles** in a Coordinate Plane Use the SSS Congruence Postulate to show that NMP DEF. SOLUTION MN = 4 and DE = 4 /

BC EF AC DF Order in the statement, Δ ABC Δ DEF, is important. 8 We Will Use CPCTE To Establish Three Types of Conclusions 1. Proving **triangles** **congruent**, like Δ ABC and Δ DEF. 2. Proving corresponding parts of **congruent** **triangles** **congruent**, like Establishing a further relationship, like A B. 9 Some Postulate Postulate 12. The SAS Postulate Every SAS correspondence is a congruency. Postulate 13. The ASA/

–4 Applies the concepts of congruency by solving problems on or off a coordinate plane; or solves problems using congruency involving problems within mathematics or across disciplines or contexts. Definitions **Congruent** **triangles**: **Triangles** that are the same size and the same shape. A B C D E F In the figure DEF ABC Congruence Statement: tells us the order in which the sides/

values for mBCA and mBCD. (2x – 16)° = 90° 2x = 106 Add 16 to both sides. x = 53 Divide both sides by 2. Example 2B: Using Corresponding Parts of **Congruent** **Triangles** Given: ∆ABC ∆DBC. Find mDBC. ∆ Sum Thm. mABC + mBCA + mA = 180° Substitute values for mBCA and mA. mABC + 90 + 49.3 = 180 m/

B E C F If ABC,DEF Right s, AB DE, AC DF, then ABC DEF. 4.5 Using **Congruent** **Triangles** Definition of **Congruent** **Triangles** (rewritten) Corresponding Parts of **Congruent** **Triangles** are **Congruent** CPCTC is used often in proofs involving **congruent** **triangles**. A is the midpoint of MT. A is the midpoint of SR. MS ll TR 1. 1. Given UR ll ST R and T are right angles/

named in order. THE ANGLE MEASURES OF A **TRIANGLE** AND **CONGRUENT** **TRIANGLES** The sum of the angle measures of a **triangle** is 180 o Example 30 o 65 o ? ? = 85 o **Congruent** **triangles** 90 o 60 o 5 cm ? ? Example **Congruent** **triangles** are **triangles** with the same shape and size Angle = 60 o ; side = 5cm Isosceles **triangles** An isosceles **triangle** is the **triangle** which has at least two sides with the same/

two sides THEOREM 4-4 If the hypotenuse and a leg of one right **triangle** are **congruent** to the corresponding parts of another right **triangle**, then the **triangles** are **congruent**. (HL) Ways to Prove Two **Triangles** **Congruent** All **triangles** – SSS, SAS, ASA, AAS Right **Triangle** - HL SECTION 4-6 Using More than One Pair of **Congruent** **Triangles** SECTION 4-7 Medians, Altitudes, and Perpendicular Bisectors Median – is the segment with endpoints/

: MCC9-12.G.SRT5, CO.7-8 Essential Question: What does it mean for two **triangles** to be **congruent** and what does CPCTC mean? **Congruent** **triangles** have **congruent** sides and **congruent** angles. The parts of **congruent** **triangles** that “match” are called corresponding parts. Essential Question: What does it mean for two **triangles** to be **congruent** and what does CPCTC mean? Complete each congruence statement. CA E D B F DEF/

same. They are just rotated or flipped in different ways. So, if all three sides of two **triangles** are the same, then the two **triangles** are **congruent**. This property is known as SSS (side-side-side). **Congruent** **Triangles** Consider the following two pairs of **triangles**. Which pair has two **congruent** **triangles**? Explain how you know. 6 8 13 6 9 14 4 9 11 4 9 A BC/

properties below will be useful in such proofs. THEOREM A B C Theorem Properties of **Congruent** **Triangles** Reflexive Property of **Congruent** **Triangles** Every **triangle** is **congruent** to itself. D E F Symmetric Property of **Congruent** **Triangles** If , then . ABC DEF J K L Transitive Property of **Congruent** **Triangles** If and , then . JKL ABC DEF Goal 2 Classwork: p. 205 #1-9 Assignment: pp. 206-9 #11-29 odd, 30-33/

included angle in one **triangle** are **congruent** to two sides and the included angle in another **triangle**, then the two **triangles** are **congruent**. **Triangle** Congruences ASA (Angle-Side-Angle) Postulate If two angles and the included side in one **triangle** are **congruent** to two angles and the included side in another **triangle**, then the two **triangles** are **congruent**. Practice In each pair below, the **triangles** are **congruent**. Tell which **triangle** congruence postulate allows you/

-26 Page 254 #6-9, 16-19, 37-40 4. 6: Congruence in Right **Triangles** 4 4.6: Congruence in Right **Triangles** 4.7: Congruence in Overlapping **Triangles** Students will be able to Prove right **triangles** are **congruent** using the Hypotenuse Leg Theorem Identify **congruent** overlapping **triangles** and use **congruent** **triangle** theorems to prove **triangles** are **congruent**. MA.912.G.4.4 andMA.912.G.4.5 and MA.912.G/

Theorems Geometry **Triangle** Congruence Theorems **Congruent** **Triangles** **Congruent** **triangles** have three **congruent** sides and and three **congruent** angles. However, **triangles** can be proved **congruent** without showing 3 pairs of **congruent** sides and angles. The **Triangle** Congruence Postulates &Theorems AAS ASA SAS SSS FOR ALL **TRIANGLES** LA HA LL HL FOR RIGHT **TRIANGLES** ONLY Theorem If two angles in one **triangle** are **congruent** to two angles in another **triangle**, the third angles must also be **congruent**. Think about/

Size SAME Shape GEOMETRY 4.3 **Congruent** Polygons have the: SAME Size SAME Shape **Congruent** **TRIANGLES** have **Congruent** CORRESPONDING SIDES **Congruent** CORRESPONDING Angles GEOMETRY 4.3 **Congruent** **TRIANGLES** have **Congruent** CORRESPONDING SIDES **Congruent** CORRESPONDING Angles A B C DE F GEOMETRY 4.3 **Congruent** **TRIANGLES** have **Congruent** CORRESPONDING SIDES **Congruent** CORRESPONDING Angles A B C DE F GEOMETRY 4.3 **Congruent** **TRIANGLES** have **Congruent** CORRESPONDING SIDES **Congruent** CORRESPONDING Angles Because CORRESPONDING parts/

E D then the 2 **triangles** are **CONGRUENT**! ***** only used with right **triangles****** The **Triangle** Congruence Postulates &Theorems LA HALL FOR RIGHT **TRIANGLES** ONLYFOR ALL **TRIANGLES** Only this one is new Summary Any **Triangle** may be proved **congruent** by: (SSS) (SAS) (ASA) (AAS) Right **Triangles** may also be proven **congruent** by HL ( Hypotenuse Leg) Parts of **triangles** may be shown to be **congruent** by **Congruent** Parts of **Congruent** **Triangles** are **Congruent** (CPCTC). Example 1 F E/

shapes sharing a side, you state that fact using the reflexive property of congruence! A C B D Draw and write down if the **triangles** are **congruent**, and by what thrmpost Proofs! The way I like to think about it to look at all the angles and sides, and don’t/B A D E C B A D E 4.5 – Using **Congruent** **Triangles** A B C D E Some Ideas that may help you. If they want you to prove something, and you see **triangles** in the picture, proving **triangles** to be **congruent** may be helpful. If they want parallel lines, look to use /

states if two angles of one **triangle** are **congruent** to two angles of another **triangle** and two corresponding non-included sides are **congruent**, then the **triangles** are **congruent**. The Angle-Angle-Side Congruence Theorem states if two angles of one **triangle** are **congruent** to two angles of another **triangle** and two corresponding non-included sides are **congruent**, then the **triangles** are **congruent**. Lesson 5.7 Proving **Triangles** **Congruent**: HL The Hypotenuse-Leg Congruence Theorem/

Presentation Holt McDougal Geometry CPCTC is an abbreviation for the phrase “Corresponding Parts of **Congruent** **Triangles** are **Congruent**.” It can be used as a justification in a proof after you have proven two **triangles** **congruent**. SSS, SAS, ASA, AAS, and HL use corresponding parts to prove **triangles** **congruent**. CPCTC uses **congruent** **triangles** to prove corresponding parts **congruent**. Remember! Example: Engineering Application A and B are on the edges of a/

sides – If the corresponding six parts of one **triangle** are **congruent** to the six parts of another **triangle**, then the **triangles** are **congruent** This is abbreviated by CPCTC (corresponding parts of **congruent** **triangles** are **congruent**) – Orientation of the **triangles** is not important. This means that the **triangles** can be flipped, slid and turned around, and if the corresponding parts are **congruent**, the **triangles** are **congruent** Segments: Angles: 1. 2. 3. Note: The order/

For example, U = F, TV = ED V = D, △ TUV △ EFD (AAS) 130° Follow-up question 3 In each of the following, name a pair of **congruent** **triangles** and give the reason. (a) A B C E F G 45° 40° 45° 5.25 cm △ ABC △ FEG (ASA) ◄ ∠ B = ∠ E, BC = EG, ∠ / corresponding sides are proportional. (i)All their corresponding angles are equal, ~is similar to △XYZ△XYZ △ABC△ABC Note: The corresponding vertices of **congruent** **triangles** should be written in the same order. If △ ABC ~ △ XYZ... 4 cm A B C 4.5 cm 40° X Y Z 2/

the properties and characteristics of polygons? Standard: MM1G1.e. Today’s Question: What does it mean for two **triangles** to be **congruent**? Standard: MM1G.3.c. Essential Question: What does it mean for two **triangles** to be **congruent** and what does CPCTC mean? **Congruent** **triangles** have **congruent** sides and **congruent** angles. The parts of **congruent** **triangles** that “match” are called corresponding parts. Essential Question: What does it mean for two/

Alexander & Koeberlein 3.1 **Congruent** **Triangles** Definition Two **triangles** are **congruent** when the six parts of the first **triangle** are **congruent** to the six parts of the second **triangle**. Definition Two **triangles** are **congruent** when the six parts of the first **triangle** are **congruent** to the six parts of the second **triangle**. What are the six parts? Definition Two **triangles** are **congruent** when the six parts of the first **triangle** are **congruent** to the six parts/

made with two straws and a 45° angle between them? If you know that two sides of a **triangle** are **congruent** to two sides of another **triangle**, what other information do you need to tell whether the **triangles** are **congruent**? Congruence Postulates Decide whether enough information is given to prove that the **triangles** are **congruent**. If there is enough information, tell which congruence postulate you would use/

- have equal lengths What did you notice about these pictures? What does it mean for two objects or figures to be **congruent**? **Congruent** figures- have the same shape and size What about **congruent** **triangles**? **Congruent** **Triangles** (same shape and size) Are these **triangles** **congruent**? Can you rotate or reflect them so that they fit on top of one another? What are the corresponding angles? A F E D/

, HL & SSS & SAS 4.6 Use **Congruent** **Triangles** CPCTC C: Corresponding P: Parts of C: **Congruent** T: **Triangles** are C: **Congruent** 4.7 Use Isosceles and Equilateral **Triangles** Base Angles Theorem: If 2 sides of a **triangle** are **congruent**, the angles opposite them are **congruent**. Converse of Base Angles Theorem: If 2 angles of a **triangle** are **congruent**, then the sides opposite them are **congruent**. Corollary to Base Angles Theorem: If a/

Agenda: 1.Do Now (10 min) 2.H-L Congruence Theorem (10 min) 3.Midsegment Theorem (30 min) 4.Isosceles **Triangles** (25 min) 5.**Congruent** **Triangles** (15 min) 6.Closure (5 min) Retake Quizzes: 10 th and 11 th graders can take retakes for any quiz/with a proof or counterexample. If two legs of a right **triangle** are the same length as two legs of another right **triangle**, then the **triangles** MUST be **congruent**. Explore Congruence of Right **Triangles** Right **triangles** consist of two legs, a hypotenuse, and a 90° angle./

the same shape and size, they are called **congruent**. We have already discussed **congruent** segments (segments with equal lengths) and **congruent** angles (angles with equal measures). **Congruent** **Triangles** **Triangles** ABC and DEF are **congruent**. If you mentally slide **triangle** ABC to the right, you can fit it exactly over **triangle** DEF by matching up the vertices. Definition of **congruent** **triangles** Two **triangles** are **congruent** if and only if their vertices can be matched/

**Congruent** **Triangles** ▫**Triangles** that are the same shape and size are **congruent**. ▫Each **triangle** has three sides and three angles. ▫If all 6 parts (3 corresponding sides and 3 corresponding angles) are **congruent**….then the **triangles** are **congruent**. Review CPCTC – Corresponding Parts of **Congruent** **Triangles** are **Congruent**/. line bisector Def. midpoint Reflexive Postulate SSS Postulate R TP X Given: Prove: Example 10: Proving **Triangles** **Congruent** Given: BC ║ AD, BC AD Prove: ∆ABD ∆CDB ReasonsStatements 5. SAS 5. /

measure of the included angle, you can construct one and only one **triangle**. CONFIDENTIAL 12 Construction **Congruent** **triangles** using SAS STEP1: Use a straightedge to draw two segments and one angle, or copy the given/ Hypothesis Conclusion ∆ABC = ∆DEF Postulate: If three sides of one **triangle** are **congruent** to three sides of another **triangle**, then the **triangles** are **congruent**. CONFIDENTIAL 26 Using SSS to prove **triangle** **Congruent** CONFIDENTIAL 27 An included angle is an angle formed by two adjacent sides/

A 42 Seg. bisector implies segments. Back STATEMENTSREASONS S S … 43 Angle bisector implies angles. Back STATEMENTSREASONS A … 44 implies right ( ) angles. Back STATEMENTSREASONS A … S 4. Given 4. 45 **Congruent** **Triangles** Proofs 1. Mark the Given and what it implies. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS, SAS, ASA) 4. List the Parts … in the order of/

Section 5.5 Objective Show corresponding parts of **congruent** **triangles** are **congruent**. Key Vocabulary - Review Corresponding parts Review: Congruence Shortcuts **Congruent** **Triangles** (CPCTC) **congruent** **triangles** cp ct c Two **triangles** are **congruent** **triangles** if and only if the corresponding parts of those **congruent** **triangles** are **congruent**. Corresponding sides are **congruent** Corresponding angles are **congruent** Example: Name the Congruence Shortcut or CBD SAS ASA SSS SSA CBD Your Turn: Name the Congruence Shortcut/

x and y that make the following **triangles** **congruent**. **Congruent** **Triangles** (CPCTC) **congruent** **triangles** cp ctc Two **triangles** are **congruent** **triangles** if and only if the corresponding parts of those **congruent** **triangles** are **congruent**. Corresponding sides are **congruent** Corresponding angles are **congruent** Congruence Statement When naming two **congruent** **triangles**, order is very important. Third Angle Theorem If two angles of one **triangle** are **congruent** to two angles of another **triangle**, then the third angles are also/

? A coordinate proof involves placing geometric figures in a coordinate plane. **Congruent** **Triangles** Which **triangles** are **congruent** by the SSS Postulate? Not **congruent** by SSS **Congruent** **Triangles** Are these **congruent** by SAS? Are these **congruent** by HL? How about: Not right **triangles**! **Congruent** **Triangles** Are these **triangles** **congruent** by ASA? Yes! No **Congruent** **Triangles** Are these **triangles** **congruent** by AAS? How about now? Yes No **Congruent** **Triangles** There are a couple of methods for organizing your thoughts when/

. ZW YX by CPCTC if ZWM YXM. Look at MWX. MW MX by the Converse of the Isosceles **Triangle** Theorem. You can prove these **triangles** **congruent** using ASA as follows: Quick Check 4-7 Using Corresponding Parts of **Congruent** **Triangles** GEOMETRY LESSON 4-7 Using Two Pairs of **Triangles** Write a paragraph proof. Given: XW YZ, XWZ and YZW are right angles. Prove: XPW YPZ Plan/

Reasons E is the midpoint of of MJ Given ME = JE Def. of Midpoint TE JE < MET = < JET TE = ________ MET = JET Proving **Triangles** **Congruent** Given: E is the midpoint of MJ; TE MJ Prove: MET = JET M J E Statements Reasons E is the midpoint of of MJ Given ME =/ JE Def. of Midpoint TE JE < MET = < JET TE = ________ MET = JET Proving **Triangles** **Congruent** Given: E is the midpoint of MJ; TE MJ Prove: MET = JET M J E Statements Reasons E is the midpoint of of MJ Given ME =/

EC by midpoint theorem. Since E is the midpoint of segment BD, segment BE is **congruent** to segment ED by midpoint theorem. Angle AEB and angle CED are vertical angles by definition. Therefore angle AEB is **congruent** to angle CED because all vertical angles are **congruent**. **Triangle** ABE is **congruent** to **triangle** CED by the side-angle-side postulate. 3. AEB and CED are vertical angles/

4.3 **Congruent** **Triangles** We will… …name and label corresponding parts of **congruent** **triangles**. …identify congruence transformations. Corresponding parts of **congruent** **triangles** **Triangles** that are the same size and shape are **congruent** **triangles**. Each **triangle** has three angles and three sides. If all six corresponding parts are **congruent**, then the **triangles** are **congruent**. Corresponding parts of **congruent** **triangles** X Z Y A C B If ΔABC is **congruent** to ΔXYZ , then vertices of the two **triangles** correspond in/

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