Understanding Angles - Blue3 Academy

Welcome to the world of basic geometry, where understanding angles is like finding the secret code to unravelling shapes and patterns around us. Whether you’re drawing shapes on paper or exploring the angles of a soccer field, knowing about it helps you make sense of how things fit together. It tells us whether lines are straight or crooked, how sharp or wide corners are, and even help us measure distances indirectly. In this article, we’ll dive into the basics of geometry, making it easy for you to see how these simple concepts are the building blocks of the world of shapes and measurements.

Watch this video to recap how angles shape our understanding of geometry. This video will guide you through the fundamental concepts that define shapes and measurements in mathematics. Get ready to see angles in action and strengthen your geometric knowledge!

Basic Geometry

In the first section of this article, we will delve into fundamental geometry concepts, focusing on angles and their properties. We will revisit essential diagrams and explore proofs central to understanding angles, such as those related to vertically opposite angles and parallel lines (e.g., l1 // l2) so as to reinforce your grasp of basic geometry principles.

Triangle Congruence and Similarity

Congruent Triangles

Congruent triangles are established when corresponding sides and angles are identical under certain conditions:

Corresponding sides are equal in length.
Corresponding angles are equal.
This congruence can be proven using different criteria such as SSS (Side-Side-Side), SAS (Side-Angle-Side), AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle), and RHS (Right angle-Hypotenuse-Side).

Similar Triangles

Similar triangles share proportional characteristics:

All corresponding angles are equal.
The ratio of corresponding sides are equal.
This similarity is established by using criteria such as AAA (Angle-Angle-Angle), SAS-similarity (Side-Angle-Side similarity), or SSS-similarity (Side-Side-Side similarity).

Two commonly encountered similar triangles are:

Pyramid triangles can be verified by the AAA test where the common, corresponding and corresponding angles are equal.
Hourglass triangles can be validated by the AAA test where the vertically opposite, alternate and alternate angles are identical.

Lastly, these are some theorems that you need to know regarding this subtopic:

AB / AD = AC / AE = BC / DE

Area of ABC / Area of ADE = (AB / AD) square

Checkpoint! Test your knowledge by answering the following question.

Check your Answers! Did you get it right?

Proof Techniques for Geometric Figures using Angles and Lines

Right Angles or Perpendicular Lines

To establish right angles or perpendicularity:

Use the Pythagoras Theorem: c=a2+b2
Use m1m2=-1.

ABCD is a Parallelogram

To prove that ABCD is a parallelogram:

Prove 2 pairs of parallel lines that are equal in length.
Prove 1 pair of parallel lines that are equal in length.
Prove the diagonals intersect at midpoint.

ABCD is a Rhombus

To prove that ABCD is a rhombus:

Prove 2 pairs of parallel lines and 4 equal sides.
Prove the diagonals intersect at midpoint and at right angles.

ABCD is a Rectangle

To prove that ABCD is a rectangle:

Prove 4 right angles.
Prove opposite angles are equal in length.

ABCD is a Square

To prove that ABCD is a square:

Prove 4 right angles.
Prove 4 equal sides.

Midpoint Theorem

In triangle ABC, the Midpoint Theorem states that:

If P and Q are the midpoints of sides AB and AC respectively,
Then PQ is parallel to BC.
Additionally, PQ is equal to half of the length of BC (and vice versa).

This theorem essentially highlights the fundamental role of midpoints in triangles by demonstrating how they relate to the parallelism and proportional division of sides, which emphasises the geometric relationship between sides and angles within triangle ABC.

The converse of the Midpoint Theorem is true as well,

If PQ is parallel to BC,
And PQ is equal to half the length of BC,
Then points P and Q are midpoints of AB and BC respectively.

Checkpoint! Test your knowledge by answering the following question.

Now, that you understand the basic concepts of the Midpoint Theorem, give the following question a go!

Check your Answers! Did you get it right?

Want to learn more?

Still want more? Then, join our interactive live teaching sessions, where we explore various topics covered in the GCE O Level Elementary Mathematics exams.