4.2 Congruence and Triangles

1. 4.2 Congruence and Triangles Geometry Fall 2005 Mrs. Spitz

2. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives: • Identify congruent figures and corresponding parts • Prove that two triangles are congruent

3. Assignment • 4.2 Worksheet A and B • Quiz 4.2 on page 210 to review for quiz next time we meet.

4. Identifying congruent figures • Two geometric figures are congruent if they have exactly the same size and shape. NOT CONGRUENT CONGRUENT

5. Corresponding angles A ≅ P B ≅ Q C ≅ R Corresponding Sides AB ≅ PQ BC ≅ QR CA ≅ RP Triangles B Q R A C P

6. Z • If Δ ABC is  to Δ XYZ, which angle is  to C?

7. Thm 4.33rd angles thm • If 2 s of one Δ are  to 2 s of another Δ, then the 3rd s are also .

8. Ex: find x ) ) 22o )) 87o )) (4x+15)o

9. Ex: continued 22+87+4x+15=180 4x+15=71 4x=56 x=14

10. 9 cm 91° 86° 113° Ex: ABCD is  to HGFE, find x and y. F E (5y-12)° G 4x – 3 cm H 4x-3=9 5y-12=113 4x=12 5y=125 x=3 y=25

11. Thm 4.4Props. of Δs A B • Reflexive prop of Δ - Every Δ is  to itself (ΔABC ΔABC). • Symmetric prop of Δ- If ΔABC ΔPQR, then ΔPQR ΔABC. • Transitive prop of Δ - If ΔABC ΔPQR & ΔPQR ΔXYZ, then ΔABC  ΔXYZ. C P Q R X Y Z

12. Start by labeling the triangles. ∆AMT can be the triangle on the left and ∆CDN can be the one on the right. Do this now. ∆AMT  ∆CDN Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides.

13. Here’s what it should look like. Next begin by labeling the drawing with tick marks to keep track of sides that are congruent. Write them down. Hint: use ∆AMT  ∆CDN Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides. A D N T M C

14. AM  MT  TA  Hint: use ∆AMT  ∆CDN Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides. A D N T M C

15. What about the angle measures? Hint: use ∆AMT  ∆CDN. The letters of the congruency statement line up. Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides. A D N T M C

16. What about the angle measures? Hint: use ∆AMT  ∆CDN. The letters of the congruency statement line up. Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides. A D N T M C A  M  T 

17. What about the angle measures? Hint: use ∆AMT  ∆CDN. The letters of the congruency statement line up. Copy the congruent triangles shown then label the vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs of congruent corresponding angles and corresponding sides. A D N T M C A  C M  D T  N

18. Some useful information • Often in proving any figures congruent, but especially triangles . . . the following theorems are especially helpful: • Reflexive Property – Segment  Segments i.e. BD  BD with shared sides of a triangle (the side in the middle). • Third Angles Theorem – when you know the other two angles are there and congruent. • Vertical Angles Theorem – the X in the triangles that look like bowties. • Last step is usually “Definition of Congruence” because this is where the congruency statement is given . . . ∆ABC  ∆DEF

19. Given: seg RP  seg MN, seg PQ  seg NQ , seg RQ  seg MQ, mP=92o and mN is 92o.Prove: ΔRQP ΔMQN N R 92o Q 92o P M

20. Statements Reasons 1. 1. given 2. mP=mN 2. subst. prop = 3. P N 3. def of s 4. RQP MQN 4. vert s thm 5. R M 5. 3rds thm 6. ΔRQP Δ MQN 6. def of  Δs