Corresponding and alternate angles

Geometry and measures

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Two parallel lines are crossed by a transversal. One of a pair of corresponding angles is $70°$. Find the other.

2 marks
One of a pair of alternate angles is $55°$. Find the alternate angle and the co-interior angle next to it.

2 marks
Two parallel lines are cut by a transversal. One angle is $65°$. Find an alternate angle and a co-interior angle.

Parallel lines cut by a transversal

3 marks
At one intersection the angle above the parallel line, on the left of the transversal, is $118°$. Find the adjacent angle on that straight line, the vertically opposite angle, and the corresponding angle on the other parallel line.

3 marks
Two parallel lines are cut by a transversal. Angles $3x$ and $2x$ are co-interior. Find $x$ and the two angles.

4 marks
An angle is $2x + 10$ and the corresponding angle is $3x - 20$. Find $x$, the corresponding angles, and a co-interior angle.

4 marks
Line $AB$ is parallel to line $CD$. Triangle $PQR$ has $P$ on $AB$ and $QR$ along $CD$. Angle $QPR = 50°$ and angle $PQR = 65°$. Find angle $PRQ$.

5 marks
Using the triangle in question $7$, explain how the parallel lines show that the triangle is isosceles, and find the angle between side $PR$ and line $CD$ on the outside of the triangle.

6 marks

Mark scheme

Answer. $70°$
- Corresponding angles are equal.
- $70°$
Answer. $55°$ and $125°$
- Alternate angles are equal.
- Co-interior angles sum to $180°$, so the other is $125°$.
Answer. $65°$ and $115°$
- The alternate angle is $65°$.
- The co-interior angle is $180 - 65 = 115°$.
Answer. $62°$, $118°$ and $118°$
- Angles on a straight line: $180 - 118 = 62°$.
- Vertically opposite angles are equal, so $118°$.
- The corresponding angle is also $118°$.
Answer. $x = 36$, angles $108°$ and $72°$
- $3x + 2x = 180$
- $x = 36$
- The angles are $108°$ and $72°$.
Answer. $x = 30$, corresponding angles $70°$, co-interior angle $110°$
- $2x + 10 = 3x - 20$, so $x = 30$.
- Each corresponding angle is $70°$.
- A co-interior angle is $110°$.
Answer. $65°$
- Angles in a triangle sum to $180°$.
- $180 - 50 - 65 = 65°$
Answer. The alternate angle to $65°$ is also $65°$, so two angles of the triangle match. The exterior angle on the straight line is $115°$.
- Angle $PQR = 65°$ and $AB$ is parallel to $CD$, so the alternate angle at $P$ on that side of the transversal is $65°$.
- The triangle has two angles of $65°$, so it is isosceles.
- Angle $PRQ = 65°$, and the adjacent angle on line $CD$ is $180 - 65 = 115°$.