060. Finding Missing Angles in Diagrams

Learning Intentions

• combine facts involving parallel lines and other geometric properties to find missing angles in a diagram

Pre-requisite Summary

• Know that corresponding angles are equal when a transversal crosses parallel lines
• Know that alternate angles are equal when a transversal crosses parallel lines
• Know that co-interior angles on parallel lines sum to ${180}^{\circ }$
• Know that vertically opposite angles are equal
• Know that angles on a straight line sum to ${180}^{\circ }$
• Know that angles around a point sum to ${360}^{\circ }$
• Know that angles in a triangle sum to ${180}^{\circ }$
• Know that a right angle is ${90}^{\circ }$ and angles in a quadrilateral sum to ${360}^{\circ }$

Worked Examples

Worked Example 1

Two parallel lines are cut by a transversal. One angle is ${68}^{\circ }$. Another angle on the same straight line is labelled $x$.
a) State the angle fact involving parallel lines.
b) State the angle fact involving a straight line.
c) Find $x$.

Worked Example 2

Two parallel lines are cut by a transversal. An angle inside the parallel lines is ${115}^{\circ }$. An adjacent angle is part of a triangle and is labelled $y$.
a) Find the angle adjacent to ${115}^{\circ }$.
b) Use the triangle angle sum if the other triangle angle is ${37}^{\circ }$.
c) Find $y$.

Worked Example 3

Two parallel lines are crossed by a transversal and one angle in the diagram is ${54}^{\circ }$. Another angle at the same intersection is vertically opposite, and a third angle at the second intersection is corresponding.
a) Find the vertically opposite angle.
b) Find the corresponding angle.
c) Explain why both are equal to ${54}^{\circ }$.

Worked Example 4

A triangle sits between two parallel lines. One exterior angle is ${127}^{\circ }$, and one interior angle of the triangle is corresponding to the adjacent interior angle at the parallel line. Another interior angle of the triangle is ${28}^{\circ }$.
a) Find the interior angle adjacent to ${127}^{\circ }$.
b) Transfer the angle using the parallel-line fact.
c) Find the third angle of the triangle.

Worked Example 5

Two parallel lines are crossed by two transversals meeting at a point. At the point, one angle is ${92}^{\circ }$, another is labelled $a$, and angles around the point must be used together with a corresponding angle of ${41}^{\circ }$.
a) Use the corresponding-angle fact to identify the ${41}^{\circ }$ angle at the point.
b) Use angles around a point.
c) Find $a$.

Worked Example 6

A quadrilateral has one pair of opposite sides parallel. Three interior angles are ${72}^{\circ }$, ${118}^{\circ }$ and $x$.
a) Identify a co-interior or interior-angle relationship from the parallel sides.
b) Use any additional quadrilateral or straight-line facts needed.
c) Find $x$.

Problems

Problem 1

Two parallel lines are cut by a transversal. One angle is ${74}^{\circ }$. Another angle on the same straight line is labelled $x$.
a) State the angle fact involving parallel lines.
b) State the angle fact involving a straight line.
c) Find $x$.

Problem 2

Two parallel lines are cut by a transversal. An angle inside the parallel lines is ${124}^{\circ }$. An adjacent angle is part of a triangle and is labelled $y$.
a) Find the angle adjacent to ${124}^{\circ }$.
b) Use the triangle angle sum if the other triangle angle is ${33}^{\circ }$.
c) Find $y$.

Problem 3

Two parallel lines are crossed by a transversal and one angle in the diagram is ${63}^{\circ }$. Another angle at the same intersection is vertically opposite, and a third angle at the second intersection is corresponding.
a) Find the vertically opposite angle.
b) Find the corresponding angle.
c) Explain why both are equal to ${63}^{\circ }$.

Problem 4

A triangle sits between two parallel lines. One exterior angle is ${136}^{\circ }$, and one interior angle of the triangle is corresponding to the adjacent interior angle at the parallel line. Another interior angle of the triangle is ${24}^{\circ }$.
a) Find the interior angle adjacent to ${136}^{\circ }$.
b) Transfer the angle using the parallel-line fact.
c) Find the third angle of the triangle.

Problem 5

Two parallel lines are crossed by two transversals meeting at a point. At the point, one angle is ${104}^{\circ }$, another is labelled $a$, and angles around the point must be used together with a corresponding angle of ${38}^{\circ }$.
a) Use the corresponding-angle fact to identify the ${38}^{\circ }$ angle at the point.
b) Use angles around a point.
c) Find $a$.

Problem 6

A quadrilateral has one pair of opposite sides parallel. Three interior angles are ${81}^{\circ }$, ${121}^{\circ }$ and $x$.
a) Identify a cointerior or interior-angle relationship from the parallel sides.
b) Use any additional quadrilateral or straight-line facts needed.
c) Find $x$.

Exercises

Understanding and Fluency

1. Two parallel lines are cut by a transversal. One angle is ${52}^{\circ }$. Find:
   a) a corresponding angle
   b) an alternate angle
   c) a cointerior angle

2. Two parallel lines are cut by a transversal. One angle is ${129}^{\circ }$. Find:
   a) a vertically opposite angle
   b) an adjacent angle on a straight line
   c) a corresponding angle

3. A triangle includes one angle of ${48}^{\circ }$ and another of ${67}^{\circ }$. Find:
   a) the third angle
   b) the supplement of the third angle
   c) whether the third angle is acute, right or obtuse

4. Angles around a point are ${85}^{\circ }$, ${146}^{\circ }$ and $x$. Find:
   a) $x$
   b) the supplement of $x$
   c) whether $x$ is acute, right, obtuse or reflex

5. Two parallel lines are cut by a transversal. One angle is ${71}^{\circ }$, and the adjacent interior angle in a triangle is needed. The second known triangle angle is ${44}^{\circ }$. Find:
   a) the angle adjacent to ${71}^{\circ }$
   b) the corresponding or alternate angle inside the triangle
   c) the third angle of the triangle

6. Two parallel lines are cut by a transversal. One exterior angle is ${117}^{\circ }$ and forms part of a quadrilateral with one other angle of ${93}^{\circ }$ and one right angle. Find:
   a) the adjacent interior angle
   b) the transferred angle inside the quadrilateral
   c) the final missing angle

7. In a diagram with parallel lines, one angle at an intersection is ${64}^{\circ }$. Another transversal forms a triangle with one other angle of ${58}^{\circ }$. Find:
   a) the angle transferred from the parallel lines
   b) the third angle of the triangle
   c) the supplement of that third angle

8. In a diagram with two parallel lines and two transversals crossing between them, one angle is ${43}^{\circ }$, another is ${97}^{\circ }$, and one angle at the central point is $x$. Find:
   a) any transferred angle equal to ${43}^{\circ }$
   b) the remaining angle around the point
   c) $x$

Reasoning

9. Explain why a missing angle in a parallel-line diagram often cannot be found using only one angle fact.

10. A student uses corresponding angles correctly, but then adds them to ${180}^{\circ }$ even though they are not on a straight line. Explain the mistake.

11. Explain why vertically opposite angles and alternate angles are different relationships, even if they can sometimes have the same size.

12. A student says that once two lines are parallel, every angle in the diagram must be equal. Explain why this is incorrect.

Problem-solving

13. Two parallel roads are crossed by a side street. One angle at the top intersection is ${66}^{\circ }$. A triangular park lies between the roads, and one other angle of the triangle is ${49}^{\circ }$. Find the third angle of the park.

14. A roof truss diagram has two parallel beams and a diagonal brace. One exterior angle is ${132}^{\circ }$, and another interior triangle angle is ${27}^{\circ }$. Find the missing angle inside the truss.

15. In a survey map, two fence lines are parallel and two paths cross between them. One transferred angle is ${58}^{\circ }$, and the angles around a meeting point also include ${121}^{\circ }$ and $x$. Find $x$.

16. A quadrilateral inside two parallel lines has one right angle, one angle of ${108}^{\circ }$, and one angle corresponding to an angle of ${72}^{\circ }$ on the upper parallel line. Find the fourth angle.

17. A bridge design shows two parallel beams cut by two supports forming a triangle. One support makes an angle of ${74}^{\circ }$ with the top beam, and the other interior angle at the base of the triangle is ${38}^{\circ }$. Find the top angle of the triangle.

18. A window frame has parallel top and bottom edges. A diagonal bar creates an angle of ${119}^{\circ }$ with the top edge. Inside the frame, a quadrilateral also includes angles of ${84}^{\circ }$ and ${90}^{\circ }$. Find the remaining interior angle.

Potential Misunderstandings

• Students may try to use only one angle fact when the diagram requires several connected steps
• Students may confuse corresponding, alternate, co-interior and vertically opposite angles
• Students may use the correct angle relationship but apply it to the wrong angle in the diagram
• Students may assume equal angles are adjacent when they are actually at different intersections
• Students may forget to use straight-line, triangle, quadrilateral or angles-around-a-point facts after transferring an angle with parallel lines
• Students may add angles that should be subtracted from ${180}^{\circ }$ or ${360}^{\circ }$
• Students may not label intermediate angles, which can make multi-step reasoning harder to follow
• Students may think diagrams are drawn to scale and guess an answer instead of using angle facts

Next: 061. Classifying and Constructing Triangles