Congruent triangles have congruent sides and congruent angles. The parts of congruent triangles that match are called corresponding parts.

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2 Congruent triangles have congruent sides and congruent angles. The parts of congruent triangles that match are called corresponding parts.

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4 A ABC DFE D B C E F AB BC DF FE A D B F C E AC DE TO PROVE TRIANGLES ARE CONGRUENT YOU DO NOT NEED TO KNOW ALL SIX

5 Before we start let s get a few things straight C Y A B X Z INCLUDED ANGLE It s stuck in between!

6 Before we start let s get a few things straight C C A B A B INCLUDED SIDE It s stuck in between!

7 Overlapping sides are congruent in each triangle by the REFLEXIVE property Vertical Angles are congruent Alt Int Angles are congruent given parallel lines

8 Side-Side-Side (SSS) Congruence Postulate All Three sides in one triangle are congruent to all three sides in the other triangle

9 Are these triangles congruent? O T C D G A If so, write the congruence statement.

10 Side-Angle-Side (SAS) Congruence Postulate Two sides and the INCLUDED angle

11 Are these triangles congruent? A A T C H If so, write the congruence statement. T

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

13 Are these triangles congruent? I E T B G O If so, write the congruence statement

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

15 Are these triangles congruent? T P O A T H If so, write a congruence statement.

16 The following slides will have pictures of triangles. You are to determine if the triangles are congruent. If they are congruent, then you should write a congruence statement and state which postulate you used to determine congruency. Δ Δ by

17 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. J K M L ΔJMK ΔLKM by SAS

18 Ex Determine 4 if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. P R S Q ΔPQS ΔPRS by SAS

19 Ex Determine 5 if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. P S Q U R ΔPQR ΔSTU by SSS T

20 Ex Determine 2 if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R S T ΔRST ΔYZX by SSS

21 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. G K I H ΔGIH ΔJIK by AAS J

22 Ex Determine 3 if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. R S T Not congruent. Not enough Information to Tell

23 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. B A C D E ΔABC ΔEDC by ASA

24 Ex Determine 6 if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent. M P R N Q Not congruent. Not enough Information to Tell

25 Warm up Are they congruent, if so, tell how AAS Not congruent 3. Not congruent

26 2-Column Proofs Going by the facts: definitions, properties, postulates, and theorems Numbering the statements and reasons Using logical order Statements Reasons

27 Given: seg WX seg. XY, seg VX seg ZX, Prove: Δ VXW Δ ZXY W Z X 1 2 V Y

28 Proof Statements Reasons 1. seg WX seg. XY 1. given seg. VX seg ZX Vertical Angles Congr Theorem 3. Δ VXW Δ ZXY 3. SAS Congr Postul

29 Given: seg RS seg RQ and seg ST seg QT Q Prove: Δ QRT Δ SRT. S R T

30 Proof Statements Reasons 1. Seg RS seg RQ 1. Given seg ST seg QT 2. Seg RT seg RT 2. Reflexive Property 3. Δ QRT Δ SRT 3. SSS Congr Postulate

31 Example Given that B C, D F, M is the midpoint of seg DF Prove Δ BDM Δ CFM B C D M F

32 Proof Statements Reasons 1. B C, D F 1. Given 2. M is the midpoint of 2. Definition of Midpt seg DF 3. Seg DM seg FM 3. Reflexive Property 4. Δ QRT Δ SRT 4. AAS Congr Theorem

33 TRIANGLE PROPORTIONALITY THEOREM Using similarity to find the missing parts of a triangle

34 TRIANGLE PROPORTIONALITY THEOREM If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. AD DB = AE EC

35 EXAMPLE 1: Find the missing side length = 18 x

36 EXAMPLE 2: Find the missing side length. x 15 = 4 10

37 EXAMPLE 3: Find the missing side length. x 15 = 2 10

38 ON YOUR OWN: 1) Find the missing side length. x 9 = 9 15

39 ON YOUR OWN: 2) Find the missing side length. 1 4 = 1 x

40 ON YOUR OWN: 3) Find the missing side length. 8 x = 14 35

41 ON YOUR OWN: 4) Find the missing side length. y 12 = 15 10

42 Properties of Parallelograms

43 Both pairs of opposite sides are parallel

44 Opposite sides are congruent

45 Opposite angles are congruent

46 onsecutive angles are supplementary

47 ABCD is a parallelogram. Find the lengths and the angle measures. 1. AD 8 B 8 C 2. m ADC E 3. m BCD 70 A 5 D

48 4. Find the value of each variable x = 5 in the parallelogram. y = y 2y 2x 6

49 5. Find the measure of D in the parallelogram. A D = 115 D x + 7 2x 1 x + 7 2x 1 B C

50 How to Prove Quadrilaterals are Parallelograms

51 How do you know if you have one?

52 1.BOTH pairs of opposite sides are parallel 2.BOTH pairs of opposite sides are congruent 3. BOTH pairs of opposite angles are congruent 4.ONE angle is supplementary to BOTH consecutive angles 5.diagonals BISECT each other 6. ONE pair of opposite sides are CONGRUENT & PARALLEL

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