Question

Find the pair of adjacent and congruent angles. The figure is on the photo attached. Please explain it to me how to find those.​ Please T _ T

Answer 1

adjacent pairs: (1, 2), (2, 4), (4, 3), (3, 1), (5, 6), (6, 8), (8, 7), (7, 5)

congruent angles: {1, 4, 5, 8}, {2, 3, 6, 7}

Explanation:

This is partly a vocabulary question.

Adjacent angles share a vertex, have a common side, and no common interior points. Adjacent angle pairs include ...

(1, 2), (2, 4), (4, 3), (3, 1), (5, 6), (6, 8), (8, 7), (7, 5)

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Congruent angles have the same measure. Where a transversal crosses parallel lines, all acute angles have the same measure, and all obtuse angles have the same measure. That is, the acute angles are congruent, and the obtuse angles are congruent.

Congruent acute angles: 1, 4, 5, 8

Congruent obtuse angles: 2, 3, 6, 7

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Additional comment

Where the lines intersect, the acute angles share a vertex and are formed from opposite rays. These are called "vertical angles." Likewise, the obtuse angles that share that vertex and are formed from opposite rays are also "vertical angles." Vertical angles at any intersection of two lines are congruent.

Answer 2

We need to find out pairs of adjacent & congruent angles in this figure. To solve that, we need to understand the meaning of the terms 'adjacent' & 'congruent' in terms of angles. So,

• Adjacent angles are those angles that share the same vertex & 1 common side. Hence, they form supplementary angles, i.e., their sum adds up to 180°.

• Congruent angles are angles that are equal to each other & have the same measure.

So, by using this definition, we can see that,

Pairs of adjacent angles ⟶ (1, 3) , (5, 7) , (2, 4) , (6, 8) , (1, 2) , (3, 4) , (5, 6) , (7, 8).

• See the attachment ⟶ In the attached figure, the red angles + yellow angles in the above given pairs form 180° & share the same vortex & 1 common side; hence, they are adjacent angles.
• Note that, although angles like the pair (3, 5), form 180°, they are not adjacent angles as they don't share the same vertex. These angles form ⟶ co-interior angles.

Pairs of congruent angles ⟶ (1, 4) , (5, 8) , (1, 8) , (1, 5) , (5, 4) , (4, 8) , (2, 6) , (3, 7) , (7, 2) , (6, 3) (2, 3) , (6, 7)

• See the attachment ⟶ In the attached figure, all the yellow angles form 1 set of congruent angles while all the red angles form the 2nd set of congruent angles.

• Here, in this figure, some congruent angles are vertically opposite angles ⟶ angles formed at a vertex when 2 lines intersect each other. (1, 4) , (5, 8) , (2, 3) , (6, 7) form vertically opposite angles & are congruent to each other.
• The other angles are formed on the basis of corresponding angles ⟶ angles in the same position but are formed at different intersections on a different set of lines.

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