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

CongruencyLast 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 ReasonsXY = XLLM = YMXM = XM< L = < Y< XMY = < XML

<LXM = < YXM

ΔLXM = ΔYXM

Prove: ΔLXM = ΔYXM~

X

YML

~

~

~

~

~

~

~

You are given this graphic and statement.Write a 2 column proof.

Given

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”.

Side Side Side

If 2 triangles have 3 corresponding pairs of sides that are congruent, then the triangles are congruent.A

B

C

AC PXAB PNCB XN∆ABC = ∆PNX~

=~=~=~

Congruency Statement

X

P

N

Given

Given

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 XPCB XN<C <X∆ABC ∆PNX

=~

=~

=~=~

Congruency Statement

Angles are INCLUDED between the congruent sides

Statements Reasons

Given

Given

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

BC

X

PN

CA XP<A <P<C <XTherefore, by ASA, ∆ABC = ∆PNX

=~=~

~

=~

Angle Side Angle

Congruency Statement

See how the side is

INCLUDED between the two angles.

There are two kinds of shortcuts

SSSSASASAAAS

HL (right triangles only)

Ones that workOnes that don’t

AAASSA

Let’s practiceWhat 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.

HG GF

<Y <S YZ ST

XZ RT

What else? 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 ♥♥

1. Duplicate AB2. Measure AC, duplicate it at A3. Measure BC, duplicate it at B4. The point of intersection is C5. 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

1st Construction

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.

1. Duplicate <A2. Duplicate AB on one ray3. Duplicate AC on the other4. 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.

1. Duplicate AB2. At vertex A, duplicate <A3. At vertex B, duplicate <B4. Name the point where the

rays of the angles intersect, C.

Your Assignment

Go to mathplease.comClick through to facebook

Add a comment under today’s discussion