I'll solve each problem on this worksheet about angle sums and exterior angles of triangles. Let me go through them one by one.
Problem 1
Find the measure of ∠q in the figure.
Looking at the diagram, I can see two parallel lines with a transversal cutting through them. Angle q is an alternate interior angle to the 130° angle shown. Since alternate interior angles are congruent when lines are parallel, ∠q = 130°.
Problem 2
∠A and ∠B are equal. ∠C exterior angle = 140°. Find the measure of ∠A and ∠B in the figure.
The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two remote interior angles. So if ∠C's exterior angle is 140°, then ∠A + ∠B = 140°.
Since ∠A and ∠B are equal, we have:
∠A + ∠A = 140°
2∠A = 140°
∠A = 70°
Therefore, ∠A = 70° and ∠B = 70°.
Problem 3
What are the measures of ∠p, ∠q, and ∠r in the figure?
In this diagram, I see two parallel lines with a transversal. The 115° angle shown is supplementary to ∠q (they form a linear pair), so:
∠q = 180° - 115° = 65°
Since ∠p and ∠q are corresponding angles (or alternate interior angles depending on how you look at it), and the lines are parallel, ∠p = ∠q = 65°.
∠r is vertically opposite to the 115° angle, so ∠r = 115°.
Therefore: ∠p = 65°, ∠q = 65°, ∠r = 115°.
Problem 4
mn and op are two parallel lines with qr as the transversal. Find the value of x.
Looking at the diagram, I can see that the angle marked "x" and the 120° angle are same-side interior angles (also called consecutive interior angles). When two parallel lines are cut by a transversal, same-side interior angles are supplementary.
So: x + 120° = 180°
x = 180° - 120°
x = 60°
Problem 5
a and b are parallel lines. Find the value of ∠m + ∠n + ∠o + ∠p.
Looking at the diagram, I can see that these four angles form a quadrilateral. The sum of the interior angles of any quadrilateral is always 360°.
Alternatively, I could think of this as two pairs of supplementary angles formed by the parallel lines and transversals, but the simplest approach is recognizing that these four angles make up the interior angles of a quadrilateral.
Therefore: ∠m + ∠n + ∠o + ∠p = 360°.
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Final Answers:
1. ∠q = 130°
2. ∠A = 70°, ∠B = 70°
3. ∠p = 65°, ∠q = 65°, ∠r = 115°
4. x = 60°
5. ∠m + ∠n + ∠o + ∠p = 360°
Parent Tip: Review the logic above to help your child master the concept of angles in triangles worksheet answers.
