Answer:
- angle list = measure
- 2, 4 = 35°
- 3, 7, 9 = 52°
- 11 = 87°
- 1, 5, 10, 12 = 93°
- 6, 8 = 128°
Step-by-step explanation:
As with a lot of math, it helps to understand the vocabulary. That helps you understand what is being said when the words are used to form a thought.
A "transversal" is a line that cuts across two parallel lines. At each intersection, 4 angles are formed. The angles are given different names, so we can talk about pairs of them being congruent.
The four angles between the parallel lines are called <em>interior</em> angles. The four angles outside the parallel lines are called <em>exterior</em> angles. When the angles are on opposite sides of the transversal, they are <em>alternate</em> angles.
In the diagram, we can identify the following pairs in each category:
- alternate interior: {3, 9}, {5, 10}
- alternate exterior: {1, 12}, {7, 52°}
When interior angles are on the same side of the transversal, they are called <em>same-side</em> or <em>consecutive</em> interior angles. Exterior angles cannot be consecutive. Here are some in that category:
- consecutive interior: {3, 6}, {5, 87°}
Angles created by a ray extending from a line are a <em>linear pair</em>. Angles of a linear pair are supplementary, that is, their sum is 180°. Angles formed by two intersecting lines, sharing only the same vertex, are called <em>vertical</em> angles. Vertical angles are both supplementary to the other angle of the linear pair of which they are a part. Since they are supplementary to the same angle, they are congruent (have the same measure). Here are some linear pairs and some vertical angles in the figure:
- linear pairs: {6, 7}, {7, 8}, {8, 9}, {6, 9}, {10, 87°}, {10, 11}, {11, 12}, {12, 87°}
- vertical angles: {1, 5}, {2, 4}, {3, 52°}, {6, 8}, {7, 9}, {10, 12}, {11, 87°}
<em>Corresponding</em> angles are ones that are in the same direction from the point of intersection. Some of those pairs are ...
- corresponding angles: {1, 10}, {5, 12}, {3, 7}, {9, 52°}
Here are the relations that help you work this problem:
- alternate interior angles are congruent
- alternate exterior angles are congruent
- vertical angles are congruent
- corresponding angles are congruent
- a linear pair is supplementary
- consecutive interior angles are supplementary
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So far, we haven't mentioned much about the angles where lines j, k, l all meet. Transversal j cuts some of the angles created by transversal k, and vice versa. So, there are some angle sum relations that also apply to corresponding angles:
- ∠1+∠2≅∠6
- ∠2+52°≅87°
- ∠3+∠4≅∠11
- ∠4+∠5≅∠8
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With an awareness of all of the above, you can figure the measures of all of the angles in the diagram.
∠1 ≅ ∠5 ≅ ∠10 ≅ ∠12 = 180° -87° = 93°
∠2+52° = 87° ⇒ ∠2 ≅ ∠4 = 87° -52° = 35°
∠3 ≅ ∠7 ≅ ∠9 ≅ 52°
∠6 ≅ ∠8 = 180° -∠7 = 128°
∠11 ≅ 87°
