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Objective 15.2

Objective 15.2 . Justify 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 180 o

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Objective 15.2

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  1. Objective 15.2 Justify congruency or similarity of polygons by using formal and informal proofs

  2. 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)

  3. Vocabulary • 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.

  4. 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

  5. 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!)

  6. Practice • Which two of the following triangles are congruent? Δ ABC Δ JIH

  7. 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

  8. Practice • Which two of the following triangles are congruent? Δ ABC Δ XZY

  9. 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

  10. Practice • Which two of the following triangles are congruent? Δ DEF Δ LKJ

  11. 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

  12. Practice • Which two of the following triangles are congruent? Δ GEF Δ SRQ

  13. More Practice • 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

  14. 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

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