Warm up No, the measures are not the same Yes, angle measures are the same and the rays go to infinity No, corresponding sides are not congruent, and we can’t tell if the angles are congruent. Yes, all corresponding sides are congruent and all corresponding angles are congruent. Yes, all corresponding parts are congruent. No, corresponding parts are not congruent. JM WC BA OP <L and < A < J and < F < M and < X < P and < B < K and < E < N and < W < O and < C

4.2 Shortcuts in Triangle Congruency Last time, we learned: IF two polygons were congruent THEN each corresponding pair of angles were congruent AND each pair of corresponding sides were congruent. Remember, in a proof, we had to LIST EACH PAIR? What postulate is that? Polygon Congruence Postulate

Statements Reasons XY = XL LM = YM XM = XM < L = < Y < XMY = < XML <LXM = < YXM ΔLXM = ΔYXM Prove: ΔLXM = ΔYXM ~ X Y M L ~ ~ ~ ~ ~ ~ ~ You are given this graphic and statement. Write a 2 column proof. Given Reflexive Property Third Angle Theorem Given All right angles are congruent Polygon Congruence Postulate Like this:

Today… You’re going to learn some shortcuts that apply to TRIANGLES ONLY. These shortcuts, if used correctly, will help you prove triangle congruency. Remember that congruency means EXACT size and shape… don’t confuse it with “similar”.

If 2 triangles have 3 corresponding pairs of sides that are congruent, then the triangles are congruent. A B C AC PX AB PN CB XN ∆ABC = ∆PNX ~ = ~ = ~ = ~ Congruency Statement X P N Given SSS

Side Angle Side If two sides and the INCLUDED ANGLE in one triangle are congruent to two sides and INCLUDED ANGLE in another triangle, then the triangles are congruent. A B C X P N CA XP CB XN <C <X ∆ABC ∆PNX = ~ = ~ = ~ = ~ Congruency Statement Angles are INCLUDED between the congruent sides StatementsReasons Given SAS

If two angles and the INCLUDED SIDE of one triangle are congruent to two angles and the INCLUDED SIDE of another triangle, the two triangles are congruent. A B C X P N CA XP <A <P <C <X Therefore, by ASA, ∆ABC = ∆PNX = ~ = ~ ~ = ~ Angle Side Angle Side Angle Congruency Statement See how the side is INCLUDED between the two angles.

There are two kinds of shortcuts SSS SAS ASA AAS HL (right triangles only) Ones that work Ones that don’t AAA SSA

Let’s practice What other information, if any, do you need to prove the two triangles congruent by SAS? Explain. To start, list the pairs of congruent, corresponding parts you already know. HGGF <Y<SYZST XZRT What else? <B = <G ~ <Z = <T ~

Get the following: ♥ 3 pieces of patty paper ♥ Ruler with centimeters ♥ Your compass ♥ pencil We are going to do 3 constructions… You have to, have to, have to, following these directions exactly. If you have a question or get stuck, please be sure to get help ♥♥ We are going to do 3 constructions… You have to, have to, have to, following these directions exactly. If you have a question or get stuck, please be sure to get help ♥♥

1.Duplicate AB 2.Measure AC, duplicate it at A 3.Measure BC, duplicate it at B 4.The point of intersection is C 5.Connect the points to create ∆ ABC On one piece of patty paper, off to the side, create 3 line segments: mBC = 4cm, mAC = 6 cm and, mAB = 8 cm 1 st Construction Construction of a triangle given three side lengths. Construction of a triangle given three side lengths.

On one piece of patty paper, off to the side, create 2 line segments and an angle: mAC = 6 cm, mAB = 8 cm, m<A = 30 ◦ 2nd Construction Construction of a triangle given 2 side lengths and an included angle. Construction of a triangle given 2 side lengths and an included angle. 1.Duplicate <A 2.Duplicate AB on one ray 3.Duplicate AC on the other 4.Connect BC

On one piece of patty paper, off to the side, create 1 line segment and 2 angles: mAB = 7 cm, m<A = 35 ◦, m<B = 50 ◦ 3rd Construction Construction of a triangle given 2 angles and an included side. Construction of a triangle given 2 angles and an included side. 1.Duplicate AB 2.At vertex A, duplicate <A 3.At vertex B, duplicate <B 4.Name the point where the rays of the angles intersect, C.

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