Geometric Reasoning
Use known angle facts to find unknown angles — including angles in triangles and quadrilaterals, supplementary angles, complementary angles, and angles on parallel lines.
Important Angle Facts
Angles on a Straight Line
Angles on a straight line add up to 180° (supplementary).
a + b = 180°
Angles at a Right Angle
Two angles that form a right angle add up to 90° (complementary).
a + b = 90°
Angles in a Triangle
The three interior angles of any triangle add up to 180°.
a + b + c = 180°
Angles in a Quadrilateral
The four interior angles of any quadrilateral add up to 360°.
a + b + c + d = 360°
Supplementary and Complementary Angles
Supplementary Angles (sum = 180°)
Two angles are supplementary when they add up to 180°. They form a straight line.
If one angle = 65°, find its supplement:
180° − 65° = 115°
Complementary Angles (sum = 90°)
Two angles are complementary when they add up to 90°. They form a right angle.
If one angle = 38°, find its complement:
90° − 38° = 52°
Vertically Opposite Angles
When two straight lines cross, the angles directly opposite each other are called vertically opposite angles, and they are always equal.
Angles and Parallel Lines
When a straight line (called a transversal) crosses two parallel lines, it creates special angle relationships.
Co-interior Angles
Between the parallel lines, same side of transversal. Add to 180°.
c + d = 180°
Alternate Angles
Between the parallel lines, opposite sides of transversal. Equal.
alternate angles are equal
Corresponding Angles
Same position at each parallel line. Equal.
corresponding angles are equal
Key Vocabulary
Supplementary Angles
Two angles that add up to 180°. Together they form a straight line.
Complementary Angles
Two angles that add up to 90°. Together they form a right angle.
Vertically Opposite
Angles formed when two lines cross. Vertically opposite angles are always equal.
Transversal
A line that crosses two or more parallel lines, creating sets of equal and supplementary angles.
Worked Examples
Two angles on a straight line are 72° and x°. Find x.
Rule: Angles on a straight line = 180°
72° + x° = 180°
x = 180° − 72°
Answer: x = 108°
A triangle has angles of 55° and 80°. Find the third angle.
Rule: Angles in a triangle = 180°
55° + 80° + x° = 180°
135° + x° = 180°
x = 180° − 135°
Answer: Third angle = 45°
A quadrilateral has angles of 95°, 88°, and 110°. Find the fourth angle.
Rule: Angles in a quadrilateral = 360°
95° + 88° + 110° + x° = 360°
293° + x° = 360°
x = 360° − 293°
Answer: Fourth angle = 67°
Knowledge Check
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Key Concepts Summary
- ●Supplementary angles add to 180° (straight line). Complementary angles add to 90° (right angle).
- ●Angles in a triangle sum to 180°. Angles in a quadrilateral sum to 360°.
- ●Vertically opposite angles are always equal when two straight lines cross.
- ●When a transversal crosses parallel lines: alternate angles are equal, corresponding angles are equal, co-interior angles sum to 180°.
- ●Always state the angle fact (reason) you are using when finding unknown angles.