Answer:
1a. HL; rotation
1b. AAS; rotation
2. 45°
3. ∠1=57°; ∠2=75°; ∠3=44°
4. x=5; y=8; z=11
Explanation:
1a. The horizontal line is a leg of each right triangle. The hypotenuse of each right triangle is marked congruent, so the two right triangles are congruent by the HL theorem. The bottom triangle is rotated 180° from the top one (about the midpoint of the horizontal segment).
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1b. The vertical angles are congruent, as are the right angles. The marked sides are congruent, but are not between the congruent angles, so the AAS postulate applies. Again, the bottom triangle is 180° rotated from the top one (about their common vertex).
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2. Exterior angle D is the sum of remote interior angles B and C, so ...
105° = 60° + ∠B
45° = ∠B
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3. ∠1 is supplementary to 123°, so is 180°-123° = 57°.
∠2 is a remote interior angle with respect to exterior angle 123°, so it plus 48° will total 123°. 123° -48° = 75° = ∠2
∠3 is the third angle in a triangle whose other two angles are 123° and 13°, so its measure is ...
180° -123° -13° = 44° = ∠3
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4. The triangles are congruent, so corresponding sides and angles are the same measure.
4x +4 = 24 ⇒ x = (24/4) -1 = 5
22 -y = 14 ⇒ y = 22 -14 = 8
(3z)° +128° +19° = 180° ⇒ z = (180 -147)/3 = 11