Answer:
A.1: ∠BAC ≅ ∠BDC ≅ ∠EDF, ∠ACD ≅ ∠ABD ≅ ∠BDE ≅ ∠CDF
A.2: ∠1 ≅ ∠4, ∠2 ≅ ∠3 ≅ ∠5 ≅ ∠6
A.3: ∠2 ≅ ∠3
B.1: ∠ACD ≅ ∠CAB, ∠CDA ≅ ∠ABC, ∠DAC ≅ ∠BCA
B.2: ∠1 ≅ ∠3 ≅ ∠5, ∠2 ≅ ∠4 ≅ ∠6
see "additional comment" regarding listing pairs
Step-by-step explanation:
There are a number of ways angles can be identified as congruent. In each case, the converse of the proposition is also true.
- opposite angles of a parallelogram are congruent
- corresponding angles where a transversal crosses parallel lines are congruent
- alternate interior angles where a transversal crosses parallel lines are congruent
- vertical angles are congruent
- any two angles with the same measure are congruent
In these exercises, pairs of angles need to be examined to see which of these relations may apply.
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A
Left
ABCD is a parallelogram, so the congruent angles are opposite angles and any that are vertical or corresponding:
∠BAC ≅ ∠BDC ≅ ∠EDF ≅ 110° (3 pairs)
∠ACD ≅ ∠ABD ≅ ∠BDE ≅ ∠CDF ≅ 70° (6 pairs)
Center
∠1 ≅ ∠4 ≅ 66° (1 pair) . . . . vertical angles
∠2 ≅ ∠3 ≅ ∠5 ≅ ∠6 ≅ 57° (6 pairs) . . . . marked with the same measure, and their vertical angles
Right
Assuming that lines appearing to go in the same direction actually do go in the same direction, the only pair of congruent angles in the figure is ...
∠2 ≅ ∠3
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B
Left
Corresponding angles in congruent triangles are congruent. Here, the congruent triangles are ΔACD ≅ ΔCAB. So, the pairs of congruent angles are ...
∠ACD ≅ ∠CAB (30°)
∠CDA ≅ ∠ABC (90°)
∠DAC ≅ ∠BCA (60°)
Right
The corresponding angles and any vertical angles are congruent. This means all the odd-numbered angles in the figure are congruent, and all the even-numbered angles in the figure are congruent. The marked 72° angles show the "horizontal" segments are parallel by the converse of the corresponding angles theorem.
∠1 ≅ ∠3 ≅ ∠5 (72°) (3 pairs)
∠2 ≅ ∠4 ≅ ∠6 (108°) (3 pairs)
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Additional comment
The question asks you to list pairs of congruent angles. When 3 things are congruent, they can be arranged in 3 pairs:
a ≅ b ≅ c ⇒ (a≅b), (a≅c), (b≅c)
Similarly, when 4 things are congruent, they can be arranged in 6 pairs:
a ≅ b ≅ c ≅ d ⇒ (a≅b), (a≅c), (a≅d), (b≅c), (b≅d), (c≅d)
In the above, we have elected not to list all of the pairs, but to list the set of congruences from which pairs can be chosen.
