o bjective 15.2 justify congruency or similarity of polygons by using formal and informal proofs
TRANSCRIPT
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