Lesson 5.1 Congruence and Triangles
Lesson 5.1 Objectives Identify congruent figures and their corresponding parts. Prove two triangles are congruent. Apply the properties of congruence to triangles.
Congruent Triangles When two triangles are congruent, then Corresponding angles are congruent. Corresponding sides are congruent. Corresponding, remember, means that objects are in the same location. So you must verify that when the triangles are drawn in the same way, what pieces match up?
Naming Congruent Parts Be sure to pay attention to the proper notation when naming parts. ABC DEF TThis is called a congruence statement. A B C D E F A D B E C F and AB DE BC EF AC DF
Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Prove Triangles are Congruent In order to prove that two triangles are congruent, we must Show that ALL corresponding angles are congruent, and Show that ALL corresponding sides are congruent. We must show all 6 are congruent!
Side-Side-Side Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
Side-Angle-Side 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.
Angle-Side-Angle Congruence If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
Angle-Angle-Side Congruence If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.
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 right triangle, then the two triangles are congruent. Abbreviate using HL
Tuesday’s Schedule Collect signed syllabi Correct lesson 5.1 day 1 assignment Review lesson 5.1 assignment Lesson 5.1 day 2 Lesson 5.1 day 2 assignment
Lesson 5.1 Day 2 Which postulate or theorem to use??
Which postulates/theorems can be used to prove triangle congruence? SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-angle) AAS (angle-angle-side) HL (hypotenuse-leg) HL can only be used in right triangles!!
Decide whether or not the congruence statement is true. Explain your reasoning! Reflexive Property of Congruence The statement is true because of SSS Congruence The statement is not true because the vertices are out of order. Because the segment is shared between two triangles, and yet it is the same segment
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem. Reflexive Property of Congruence SAS Congruence No
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem. Reflexive Property of Congruence Yes they are congruent! HL Reflexive Property of Congruence Not congruent
Decide whether or not there is enough information to conclude triangle congruence. If so, state the postulate or theorem. Reflexive Property of Congruence Yes they are congruent! ASA Reflexive Property of Congruence Yes they are congruent! AAS
Wednesday Collect signed syllabi Correct/review 5.1 day 2 Notes over lesson 5.2 Assignment 5.2
Lesson 5.2 Proving Triangles are Congruent
Review What does congruent mean? Draw two triangles that appear to be congruent. Label your drawings to make the two triangles congruent.
Complete the proof If 2(x+12)=90, then x= (x+12)=901. Given 2. 2x+24=902. Distributive Property 3. 2x=663. SPOE 4. X=334. DPOE
Complete the proof Given Reflexive POC SSS Congruence
Complete the proof Given: Prove: 1.1.Given Given 3.3. AIA 4.4. Reflexive 5.4. SAS
Construct a proof Definition of midpoint 2. Definition of midpoint 3. Vertical angles 4. SAS
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 be written as a:b.a:b. A A ratio is a fraction, so the denominator cannot be zero. Ratios should always be written in simplified form. 5 / 10 1/21/2
Proportional If two ratios are equal after they are simplified, then they are said to be proportional. These two ratios are proportional.
Similarity of Trianlges Two Triangles are similar when the following two conditions exist Corresponding angles are congruent Correspondng sides are proportional Means that all side fit the same ratio. The symbol for similarity is ~ ABC ~ FGH This is called a similarity statement.
Scale Factor Since all the ratios should be equivalent to each other, they form what is called the scale factor. We represent scale factor with the letter k. This is most easily found by find the ratio of one pair of corresponding side lengths. Be sure you know the polygons are similar. k = 20 / 5 k =
Angle-Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Theorem 8.2: Side-Side-Side Similarity If the corresponding sides of two triangles are proportional, then the triangles are similar. Your job is to verify that all corresponding sides fit the same exact ratio!
Theorem 8.3: Side-Angle-Side Similarity If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Your task is to verify that two sides fit the same exact ratio and the angles between those two sides are congruent!
Using Theorems…which one do I use? These theorems share the abbreviations with those from proving triangles congruent. SSS SAS So you now must be more specific SSS Congruence SSS Similarity SAS Congruence SAS Similarity You chose based on what are you trying to show? Congruence Similarity