Plane Geometry: Theorems

Grade 12 Math - Plane Geometry: Theorems

Lesson Objectives

Recall and apply key theorems in plane geometry
Use geometrical reasoning to prove statements about angles and triangles
Apply theorems to solve geometric problems involving parallel lines, triangles, and circles
Interpret given diagrams and construct logical geometric proofs

Lesson Introduction

Plane Geometry involves properties and relationships between points, lines, angles, and shapes in a flat (2D) space. This lesson focuses on applying geometric theorems logically and accurately, especially for angles in triangles, parallel lines, and circle geometry.

Core Lesson Content

Angle Sum Theorem in a Triangle

The sum of the angles in any triangle is $180^\circ$

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two opposite interior angles.

Isosceles Triangle Theorem

If two sides of a triangle are equal, the angles opposite them are equal.

Corresponding, Alternate and Co-interior Angles

Formed when a transversal crosses parallel lines:

Corresponding angles are equal
Alternate angles are equal
Co-interior angles are supplementary

Circle Theorems

Angle at the center is twice the angle at the circumference
Angles in the same segment are equal
Angle in a semicircle is $90^\circ$
Opposite angles of a cyclic quadrilateral are supplementary

Worked Examples

Example 1: In

\triangle ABC

, if

\angle A = 60^\circ

and

\angle B = 80^\circ

, find

\angle C

.
Solution:

\angle C = 180^\circ - (60^\circ + 80^\circ) = 40^\circ

Example 2: Find the value of an exterior angle of a triangle if the opposite interior angles are

65^\circ

and

45^\circ

.
Solution: Exterior angle =

65^\circ + 45^\circ = 110^\circ

Example 3: In an isosceles triangle, two angles are

70^\circ

. Find the third angle.
Solution:

180^\circ - (70^\circ + 70^\circ) = 40^\circ

Example 4: Given two parallel lines cut by a transversal, one angle is

110^\circ

. Find its corresponding angle.
Solution: Corresponding angle =

110^\circ

(equal)

Example 5: Find the alternate angle of

65^\circ

formed by a transversal.
Solution: Alternate angle =

65^\circ

Example 6: In a circle, an angle at the center is

100^\circ

. Find the angle at the circumference on the same arc.
Solution:

100^\circ \div 2 = 50^\circ

Example 7: In a cyclic quadrilateral, one angle is

120^\circ

. Find its opposite angle.
Solution:

180^\circ - 120^\circ = 60^\circ

Example 8: In a triangle, one side is extended forming an exterior angle of

130^\circ

. Find the interior opposite angles if one is

70^\circ

.
Solution: Other interior angle =

130^\circ - 70^\circ = 60^\circ

Example 9: Find the angle in a semicircle.
Solution: Always

90^\circ

Example 10: Prove that angles in the same segment are equal.
Solution: By construction and congruency of arcs, they subtend the same angle on the circumference.

Exercises

In triangle XYZ, $\angle X = 50^\circ$ , $\angle Y = 60^\circ$ . Find $\angle Z$ .
[WAEC] Find the exterior angle of a triangle if two interior opposite angles are $40^\circ$ and $55^\circ<span class="past-question">[Past Question]</span>.</li> <li><span class="past-question">[NECO]</span> In an isosceles triangle, the unequal angle is 36^\circ$ . Find the base angles. [Past Question]
If a transversal makes an angle of $75^\circ$ on one line, find the corresponding angle on the other line.
Two lines are parallel and cut by a transversal. One alternate angle is $85^\circ$ . Find the other.
[NECO] In a circle, the angle subtended at the center is $120^\circ$ . Find the angle at the circumference.[Past Question]
In a cyclic quadrilateral, one angle is $97^\circ$ . Find its opposite angle.
[WAEC] The angle in a semicircle is always _____. Justify your answer. [Past Question]
Find the value of the interior opposite angles if the exterior angle is $150^\circ$ and one interior is $65^\circ$ .
Prove that corresponding angles are equal when a transversal crosses two parallel lines.

Conclusion/Recap

Theorems in plane geometry provide logical rules that help us solve problems with confidence. A strong understanding of these theorems enables students to justify each step and construct accurate geometric arguments, critical for both theoretical and practical applications.