Mathematics 7th grade | UNIT 5: Angles 5.4 Solving angle problems
This diagram shows two straight lines crossing.
Angles labelled ao and care vertically opposite angles. Angles labelled bo and do are also opposite angles.
You can prove that vertically opposite angles are equal, as follows.
• $a + d = 180$ because they are angles on a straight line. Therefore $a = 180 - d$.
• $c + d = 180$ because they are angles on a straight line. Therefore $c = 180 - d$.
• Since a and care both equal to 180 - d, this means that $a = c$.
In the same way, you can show that $b= d$.
A special case of this is when two lines are perpendicular.
All the angles are ${90^ \circ }$.
Many equal angles are created when two parallel lines are crossed by a third line.
The arrows drawn on the diagram show that PQ and RS are parallel. XY is a straight line.
Check the following facts.
• $a + b = 180$ They are angles on a straight line.
• $a = c$ and $b= d$ They are opposite angles.
• $a = e$ and $b = f$ This is because PQ and RS are parallel.
• $a = c = e = g$
• $b = d = f = h$
Exercise 5.4
1) Prove angle $APC =$ angle DPB.
2) Three straight lines cross at one point.
Calculate the values of a, b, cand d. Give reasons for your answers.
3) Lines WX and YZ are parallel.
One angle of ${77^ \circ }$ is marked. Find the values of a, band c.
4) This diagram shows four identical triangles.
Look at the angles at point A.
Explain why this shows that the angles of a triangle add up to ${180^ \circ }$.
5) ABC is an isosceles triangle. $AB = BC$ and angle $BAC = {40^ \circ }$.
An isosceles triangle has two equal sides and two equal angles.
Calculate the other two angles of the triangle.
6) Calculate the value of a.
7) Explain why AB and AC are equal in length.
