OBJECTIVE 15.2Justify congruency or similarity of polygons by using formal and informal proofs

VOCABULARY

Linear pair – two angles that share a side and form a line. The measures of these angles add up to 180o

Vertical angles are the angles opposite each other when two lines cross. Vertical angles are congruent (ao = bo)

Included sides are sides that are in between two angles that are being referenced. If we are talking about angles A & B, side c would be an included side.

Included angles are angles that are in between two sides that are being referenced. If we are talking about sides b and c, angle A would be an included angle.

VOCABULARY

CONGRUENT TRIANGLES

Two triangles are considered congruent when all 3 corresponding angles are congruent and all 3 corresponding sides are congruent

However, you don’t always need to know all 6 of those measurements to prove a triangle is congruent.

There are 4 congruency shortcuts you can use to prove that two triangles are congruent

SIDE-SIDE-SIDE (SSS)

The first congruency shortcut is side-side-side (SSS)

If all three corresponding sides of two triangles are congruent, then the two triangles are congruent.

If a = n, b = l, and c = m, then A corresponds to N, B corresponds to L and C corresponds to M. Thus,

ΔABC ΔNLM (the order here is VERY important!)

PRACTICE

Which two of the following triangles are congruent?

Δ ABC Δ JIH

SIDE-ANGLE-SIDE (SAS)

The second congruency shortcut is side-angle-side (SAS).

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

Δ ABC ΔLOM

PRACTICE

Which two of the following triangles are congruent?

Δ ABC Δ XZY

ANGLE-SIDE-ANGLE (ASA)

The third congruency shortcut is angle-side-angle (ASA).

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Δ ABC ΔZYX

PRACTICE

Which two of the following triangles are congruent?

Δ DEF Δ LKJ

ANGLE-ANGLE-SIDE (AAS)

The final congruency shortcut is angle-angle-side (AAS).

If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

Δ ABC ΔQSR

PRACTICE

Which two of the following triangles are congruent?

Δ GEF Δ SRQ

Sometimes you’ll be given some information about triangles and line segments and will have to pull out information about congruency.

Since M is the midpoint of AB and PQ, we know that: PM = QM MA = MB.

This means we have 2 congruent sides. We could use SSS or SAS.

We don’t know anything about PA and BQ, but what about the included angles, 1 & 2?

Well, they’re a vertical pair! So angle 1 = angle 2 and we can use SAS to say that ΔAPM ΔBQM

MORE PRACTICE

SHARED SIDES If two triangles share a side, then that side is

equal to itself and can be used as a congruent side:

So LX = LX, angle NLX = angle XLM and right angles are congruent as well. So we can use ASA to say that ΔNLX ΔMLX