O BJECTIVE 15.2 Justify congruency or similarity of polygons by using formal and informal proofs.
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Transcript of O BJECTIVE 15.2 Justify congruency or similarity of polygons by using formal and informal proofs.
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