Applying Triangle Sum Properties Section 4.1

Triangles Triangles are polygons with three sides. There are several types of triangle: Scalene Isosceles Equilateral Equiangular Obtuse Acute Right

Scalene Triangles Scalene triangles do not have any congruent sides. In other words, no side has the same length. 6cm 3cm 8cm

Isosceles Triangle A triangle with 2 congruent sides. 2 sides of the triangle will have the same length. 2 of the angles will also have the same angle measure.

Equilateral Triangles All sides have the same length

Equiangular Triangles All angles have the same angle measure.

Acute Triangle All angles are acute angles.

Right Triangle Will have one right angle.

Obtuse Angle Will have one obtuse angle.

Exterior Angles vs. Interior Angles Exterior Angles are angles that are on the outside of a figure. Interior Angles are angles on the inside of a figure.

Interior or Exterior?

Interior or Exterior?

Interior or Exterior?

Triangle Sum Theorem (Postulate Sheet) States that the sum of the interior angles is 180. We will do algebraic problems using this theorem. The sum of the angles is 180, so x + 3x + 56= 180 4x + 56= 180 4x = 124 x = 31

Find the Value for X 2x + 15 3x 2x + 15 + 3x + 90 = 180 5x + 105 = 180

Corollary to the Triangle Sum Theorem (Postulate Sheet) Acute angles of a right triangle are complementary. 3x + 10 5x +16

Exterior Angle Sum Theorem The measure of the exterior angle of a triangle is equal to the sum of the non-adjacent interior angles of the triangle

88 + 70 = y 158 = y

2x + 40 = x + 72 2x = x + 32 x = 32

Find x and y 3x + 13 46o 8x - 1 2yo

To define congruent triangles To write a congruent statement 4.1 Apply Congruence and Triangles 4.2 Prove Triangles Congruent by SSS, SAS Objectives: To define congruent triangles To write a congruent statement To prove triangles congruent by SSS, SAS

Congruent Polygons

Congruent Triangles (CPCTC) Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.

Congruence Statement When naming two congruent triangles, order is very important.

Example Which polygon is congruent to ABCDE? ABCDE  -?-

Properties of Congruent Triangles

Example What is the relationship between C and F?

Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Congruent Triangles Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

Congruence Shortcuts? Will one pair of congruent sides be sufficient? One pair of angles?

Congruence Shortcuts? Will two congruent parts be sufficient?

Will three congruent parts be sufficient? And if so….what three parts? Congruent Shortcuts? Will three congruent parts be sufficient? And if so….what three parts?

Section 4.3 Proving Triangles are Congruents by SSS

Draw any triangle using any 3 size lines For me I use lines of 5, 4, and 3 cm’s. Now use the same lengths and see if you can make a different triangle. Now measure both triangles angles and see what you get. 3cm 53 53 90 5cm 3cm 4cm 5cm 90 37 4cm 37

Are the following triangles congruent? Why? 10 6 6 6 6 YES, all sides are equal so SSS a. 10 9 10 8 10 No, all sides are not equal 8 ≠ 6, so fails SSS b. 6 9

Use the SSS Congruence Postulate Decide whether the congruence statement is true. Explain your reasoning. SOLUTION Given Given Reflexive Property So, by the SSS Congruence Postulate,

4.4:Prove Triangles Congruent by SAS and HL Goal:Use sides and angles to prove congruence.

Vocabulary Leg of a right triangle: In a right triangle, a side adjacent to the right angle is called a leg. Hypotenuse:In a right triangle, the side opposite the right angle is called the hypotenuse. Hypotenuse Leg

4.5 ASA and AAS

Before we start…let’s get a few things straight C X Z Y INCLUDED SIDE

Angle-Side-Angle (ASA) Congruence Postulate Two angles and the INCLUDED side

Angle-Angle-Side (AAS) Congruence Postulate Two Angles and One Side that is NOT included

Your Only Ways To Prove Triangles Are Congruent SSS SAS ASA AAS NO BAD WORDS Your Only Ways To Prove Triangles Are Congruent

Alt Int Angles are congruent given parallel lines Things you can mark on a triangle when they aren’t marked. Overlapping sides are congruent in each triangle by the REFLEXIVE property Alt Int Angles are congruent given parallel lines Vertical Angles are congruent

Ex 1 DEF NLM

Ex 2 What other pair of angles needs to be marked so that the two triangles are congruent by AAS? F D E M L N

Ex 3 What other pair of angles needs to be marked so that the two triangles are congruent by ASA? F D E M L N

ΔGIH  ΔJIK by AAS Ex 4 G I H J K Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 4 G I H J K ΔGIH  ΔJIK by AAS

ΔABC  ΔEDC by ASA B A C E D Ex 5 Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. B A C E D Ex 5 ΔABC  ΔEDC by ASA

ΔACB  ΔECD by SAS Ex 6 E A C B D Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 6 E A C B D ΔACB  ΔECD by SAS

ΔJMK  ΔLKM by SAS or ASA Ex 7 J K L M Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 7 J K L M ΔJMK  ΔLKM by SAS or ASA

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. Ex 8 J T L K V U Not possible

Postulate 20:Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Example 1:Use the SAS Congruence Postulate Write a proof.

Example 2:Use SAS and properties of shapes

Checkpoint

Checkpoint

Theorem 4.5:Hypotenuse-Leg Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruent.

Example 3:Use the Hypotenuse-Leg Theorem Write a proof.

Example 3:Use the Hypotenuse-Leg Theorem

Example 3:Use the Hypotenuse-Leg Theorem

Example 4:Choose a postulate or theorem

Example 4:Choose a postulate or theorem

Using Congruent Triangles: CPCTC Academic Geometry

Proving Parts of Triangles Congruent You know how to use SSS, SAS, ASA, and AAS to show that the triangles are congruent. Once you have triangles congruent, you can make conclusions about their other parts because, by definition, corresponding parts of congruent triangles are congruent. Abbreviated CPCTC

Proving Parts of Triangles Congruent In an umbrella frame, the stretchers are congruent and they open to angles of equal measure. Given SL congruent to SR <1 congruent <2 Prove that the angles formed by the shaft and the ribs are congruent shaft stretcher rib l r 3 4 1 2 c s

Proving Parts of Triangles Congruent Prove <3 congruent <4 Statement Reason shaft stretcher rib l r 3 4 1 2 c s

Proving Parts of Triangles Congruent Given <Q congruent <R <QPS congruent <RSP Prove SQ congruent PR Statements Reasons r p q s

Proving Parts of Triangles Congruent Given <DEG and < DEF are right angles. <EDG congruent <EDF Prove EF congruent EG Statements Reasons d e f g

4.7 Isosceles and Equilateral Triangles Chapter 4 Congruent Triangles

4.5 Isosceles and Equilateral Triangles Isosceles Triangle: Vertex Angle Leg Leg Base Angles Base *The Base Angles are Congruent*

Isosceles Triangles Theorem 4-3 Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent B <A = <C A C

Isosceles Triangles Theorem 4-4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent B Given: <A = <C Conclude: AB = CB A C

Isosceles Triangles Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base B Given: <ABD = <CBD Conclude: AD = DC and BD is ┴ to AC A C D

Equilateral Triangles Corollary: Statement that immediately follows a theorem Corollary to Theorem 4-3: If a triangle is equilateral, then the triangle is equiangular Corollary to Theorem 4-4: If a triangle is equiangular, then the triangle is equilateral

Using Isosceles Triangle Theorems Explain why ΔRST is isosceles. T U Given: <R = <WVS, VW = SW Prove: ΔRST is isosceles R W Statement Reason V 1. VW = SW 1. Given S 2. m<WVS = m<S 2. Isosceles Triangle Thm. 3. m<R = m<WVS 3. Given 4. m<S = m<R 4. Transitive Property 5. ΔRST is isosceles 5. Def Isosceles Triangle

Using Algebra Find the values of x and y: M ) ) ΔLMN is isosceles 27° y° y° m<L = m< N = 63 m<LM0 = y = m<NMO 63° N 63 + 63 + y + y = 180 x° 126 + 2y = 180 63° O - 126 -126 L 2y = 54 2 2 27 + 63 + x = 180 y = 27 90 + x = 180 -90 -90 x = 90

Landscaping A landscaper uses rectangles and equilateral triangles for the path around the hexagonal garden. Find the value of x. x°