The angles of a triangle add up to \(180^{\circ}\), and an exterior angle equals the sum of the two remote interior angles. When parallel lines are cut by a transversal, corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to \(180^{\circ}\).
Key facts:
- Standard 8.G.A.5: angle sum, exterior angles, parallel lines with a transversal, and the AA criterion.
- Triangle angle sum: \(180^{\circ}\).
- Exterior angle = sum of the two remote interior angles.
- Parallel lines: corresponding, alternate interior and alternate exterior angles are equal; same-side interior angles sum to \(180^{\circ}\).
- AA criterion: two equal pairs of angles make triangles similar.
What is the angle sum of a triangle?
The three interior angles of every triangle add up to \(180^{\circ}\). To see why, draw a line through one vertex parallel to the opposite side. The two new angles beside the vertex equal the other two interior angles, because they are alternate interior angles. Together with the third angle they form a straight line, which is \(180^{\circ}\).
Example. A triangle has angles \(52^{\circ}\) and \(71^{\circ}\). The third angle is \(180-52-71=57\), so it measures \(57^{\circ}\).
What is the exterior angle theorem?
An exterior angle forms when one side of a triangle is extended. It equals the sum of the two remote interior angles, the interior angles that are not next to it. The reason: the exterior angle and its neighbor make \(180^{\circ}\), and the three interior angles also make \(180^{\circ}\). In the figure, \(45+70=115\).
What angles form when a transversal crosses parallel lines?
Transversal: a line that crosses two or more other lines.
Corresponding angles: angles in the same position at each intersection.
Alternate interior angles: between the lines, on opposite sides of the transversal.
Alternate exterior angles: outside the lines, on opposite sides of the transversal.
Same-side interior angles: between the lines, on the same side of the transversal.
When the two lines are parallel, corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles add to \(180^{\circ}\).
| Pair type | Example in the figure | Relationship |
|---|---|---|
| Corresponding | \(\angle2\) and \(\angle6\) | equal |
| Alternate interior | \(\angle3\) and \(\angle5\) | equal |
| Alternate exterior | \(\angle1\) and \(\angle7\) | equal |
| Same-side interior | \(\angle3\) and \(\angle6\) | sum is \(180^{\circ}\) |
How do you find a missing angle?
- Name the relationship between the angles.
- Write an equation: equal angles, or a sum of \(180^{\circ}\).
- Solve for the variable and find each angle.
- Check that the angles fit the figure.
Worked example. Alternate interior angles measure \((3x+5)^{\circ}\) and \(80^{\circ}\). They are equal, so \(3x+5=80\), \(3x=75\) and \(x=25\). Check: \(3(25)+5=80\).
How do angles show that triangles are similar?
The angle-angle (AA) criterion says that if two angles of one triangle equal two angles of another, the triangles are similar. The third angles must match too, because each triangle sums to \(180^{\circ}\). For example, a triangle with \(50^{\circ}\) and \(60^{\circ}\) angles and a triangle with \(60^{\circ}\) and \(70^{\circ}\) angles both have angles \(50^{\circ}\), \(60^{\circ}\), \(70^{\circ}\), so they are similar and their sides are proportional.
Angles, Parallel Lines and Triangles: frequently asked questions
What is the sum of the angles in a triangle?
The three interior angles of any triangle add up to \(180^{\circ}\). If two angles are \(48^{\circ}\) and \(67^{\circ}\), the third is \(180-48-67=65\), so it measures \(65^{\circ}\).
What is the exterior angle theorem?
An exterior angle of a triangle equals the sum of the two remote interior angles. If the remote angles are \(50^{\circ}\) and \(65^{\circ}\), the exterior angle is \(115^{\circ}\).
Which angles are equal when parallel lines are cut by a transversal?
Corresponding angles, alternate interior angles and alternate exterior angles are equal. Same-side interior angles are not equal in general: they add to \(180^{\circ}\).
How can you tell if two lines are parallel using angles?
If a transversal makes equal corresponding angles, or equal alternate interior angles, the lines are parallel. Same-side interior angles that add to \(180^{\circ}\) also show the lines are parallel.
Why do two equal angles make triangles similar?
If two angles match, the third angles match too, because the angles of each triangle sum to \(180^{\circ}\). All three pairs of angles are equal, so one triangle is an enlargement or reduction of the other. This is the AA criterion.
See every chapter on the Grade 8 math page.