13. Congruency and Angle Properties

In this checkpoint, we’re going to go through congruence, transformations and quadrilaterals. While they’re long sounding words, the concepts are important in geometry. We’ll do the following:

Understand and apply the concept of congruence.
Use transformations (translations, rotations, reflections) to determine congruence.
Create patterns and tessellations using congruent shapes.
Identify and apply the congruence conditions for triangles (SSS, SAS, ASA, RHS).
Establish and apply properties of quadrilaterals using congruent triangles and angle properties.
Solve numerical problems related to congruence and quadrilaterals.

So, let’s get started.

Congruence and Transformations

Two shapes are congruent if they have the same size and shape. This means all corresponding sides and angles are equal. We use the symbol ≅ to denote congruence.

For example, two identical squares, two circles with the same radius, two equilateral triangles with the same side length are congruent.

The following are not congruent: a square and a rectangle (different shapes), a small circle and a large circle (different sizes), a scalene triangle and an equilateral triangle.

Let’s do some quick practice.

Which of the following pairs of shapes are definitely congruent?

Draw two congruent rectangles.

If △ABC≅△DEF, and $AB = 5cm, what is the length of DE?

If △PQR≅△XYZ, and ∠P=70∘, what is the measure of ∠X?

Now, let’s look at transformations.

Transformations are ways to move shapes on a plane without changing their size or shape i.e. they preserve congruence – this means that the transformed shape is congruent to the original shape as the only thing that has changed is position. We'll focus on three types:

Translations: "Sliding" a shape without rotating or flipping it. We describe a translation by how far the shape moves horizontally and vertically. Example: Translate a triangle 3 units to the right and 2 units up.
Rotations: "Turning" a shape around a fixed point (the centre of rotation) by a certain angle. Example: Rotate a square 90 degrees clockwise around its centre.
Reflections: "Flipping" a shape over a line (the line of reflection). Example: Reflect a triangle over the x-axis.

Let’s do some simple practice – watch the video for solutions:

Draw a triangle and then translate it 4 units to the left and 1 unit down.
Draw a square and then rotate it 90 degrees counterclockwise around one of its vertices.
Draw a simple shape and then reflect it over a vertical line.
If you translate a shape, is the new shape congruent to the original shape? Why or why not?

Tessellations

A tessellation is a pattern of shapes that fit together perfectly without any gaps or overlaps. Congruent shapes can be used to create tessellations. Transformations are used to position the shapes.

There are different types of tessellations:

Regular Tessellations: Made up of only one type of regular polygon (e.g., squares, equilateral triangles, regular hexagons).
Semi-Regular Tessellations: Made up of more than one type of regular polygon, with the same arrangement of polygons at each vertex.

Which of the following regular polygons can tessellate on their own: square, equilateral triangle, pentagon, hexagon? Watch the video! (The interior angles of a regular pentagon [108 degrees] do not divide evenly into 360 degrees, so they cannot meet at a vertex without gaps or overlaps.)

Congruence of Triangles

The following are 4 congruence conditions of triangles:

SSS(Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

SAS(Side-Angle-Side):If two sides and the included angle (the angle between those sides) of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

ASA(Angle-Side-Angle):If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

RHS(Right angle-Hypotenuse-Side): If the hypotenuse and one other side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, the two triangles are congruent.

Let’s do some quick questions:

If △ABC≅△DEF, AB=DE, BC=EF, and AC=DF, which congruence condition proves this?
If △PQR≅△XYZ, ∠P=∠X, PQ=XY, and ∠Q=∠Y, which congruence condition proves this?
Why is SSA not a congruence condition?

Solutions: SSS, ASA

Quadrilaterals

A quadrilateral is a four-sided polygon with internal angles totalling 360 degrees. Here are some examples:

Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Diagonals bisect each other.
Rectangle: A parallelogram with four right angles. Diagonals are equal.
Square: A rectangle with four equal sides.
Rhombus: A parallelogram with four equal sides. Diagonals are perpendicular bisectors of each other.
Trapezium (Trapezoid): One pair of opposite sides is parallel.
Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular. One diagonal bisects the other.

Now that we know about triangles and quadrilaterals, how do they relate and why are they useful?

Let’s move onto the next section:

Using Congruent Triangles to Prove Properties

We have a set of tools (the triangle congruence rules: SSS, SAS, ASA, RHS) that we know are reliable. We use these tools to build our understanding of more complicated shapes. We're not just stating that a parallelogram has opposite sides equal; we're showing why it's true, based on solid logic.

We know that an interior angle is simple an angle INSIDE a shape.

Triangle: A triangle has three interior angles. The sum of the interior angles of any triangle is always 180 degrees. This is a fundamental fact.
Quadrilateral: A quadrilateral has four interior angles. The sum of the interior angles of any quadrilateral is always 360 degrees. You can see this by dividing any quadrilateral into two triangles with a diagonal; each triangle contributes 180 degrees.

Let's look at types of angles and what they look like.

Orias / CC BY-SA (https://creativecommons.org/licenses/by-sa/3.0). Built upon by Exam Success.

Important tips to remember when dealing with angles are:

Right angle – lines that intersect are perpendicular. A little square represents a right angle.
Straight angle – Angle in a straight line and is 180 degrees (2 sides of the straight line add to 360 degrees).
Angles around a point add up to 360 degrees.
Angles in a square add up to 360 degrees.
Angles in a triangle add up to 180 degrees.
Complementary angles are angles that together add up to 90°.
Supplementary angles are angles that together add up to 180°.

Interestingly, there are some rules with angles when parallel lines cross. These are:

Corresponding angles are equal
Alternate interior angles are equal
Same side interior angles add to 180° if on a straight line.

Let’s prove that a parallelogram has opposite side that are equal using the triangle rules. We can do that by:

Splitting the parallelogram into two triangles.
Apply triangle rule congruency by ASA
Therefore, corresponding parts of congruent triangle are equal.

Multi-Part Questions

Now, that we know these concepts, it’s time to apply them to a scenario-based question.

Question 1: Trapezium Roof

A trapezium-shaped roof panel is being designed. The two bottom angles of the trapezium are each 60°, while the two top angles are unknown.

The builder wants to determine the measure of the two missing angles.

Using your knowledge of quadrilateral properties, calculate the value of the two missing angles.

Solution

You should recall that the sum of all interior angles in any quadrilateral is 360 degrees. Then set up your formula as follows:

Solving for x would give you 120 degrees.

Question 2 – The Metal Frame

A construction engineer is designing a parallelogram-shaped metal frame ABCD for a billboard. A diagonal support beam AC is placed from one corner to the opposite corner, dividing the parallelogram into two triangles, △ABC and △CDA.

The engineer measures:

One pair of opposite sides are equal: AB=CD=3x+4
The other pair of opposite sides are equal: BC=AD=2x+10
The diagonal AC is shared between both triangles.

To ensure the frame is structurally stable, the engineer needs the two triangles created by the diagonal to be congruent. Additionally, the engineer wants the parallelogram to have all sides equal (forming a rhombus) for optimal stability.

Using triangle congruence rules, prove that △ABC ≅ △CDA.
Find the value of x that makes all sides of the parallelogram equal.

Solution

Prove △ABC ≅ △CDA. We use the Side-Side-Side (SSS) congruence rule by comparing the sides of the two triangles:△ABC has sides AB, BC, and AC. △CDA has sides CD, DA, and AC. Compare the corresponding sides: AB = CD = 3x + 4 (given). BC = DA = 2x + 10 (given, noting DA = AD). AC = AC (shared diagonal). Since all three pairs of corresponding sides are equal (AB = CD, BC = DA, AC = AC), by the SSS congruence rule, △ABC ≅ △CDA.
Find the Value of x That Makes All Sides Equal. The engineer wants ABCD to be a rhombus, meaning all sides of the parallelogram must be equal: AB = CD = 3x + 4, BC = AD = 2x + 10. Set the side lengths equal to each other: 3x + 4 = 2x + 10, 3x - 2x = 10 - 4, x = 6. Verify with x = 6: AB = CD = 3(6) + 4 = 18 + 4 = 22, BC = AD = 2(6) + 10 = 12 + 10 = 22. All sides are now 22 units, so ABCD is a rhombus, a special type of parallelogram.

Practice time!

Now, it's your turn to practice.

The questions in this checkpoint are provided to give you an introduction to possible questions you may see in your exam. Don't worry too much as you'll continue to build your skills throughout the course.

Click on the button below and start your practice questions. We recommend doing untimed mode first, and then, when you're ready, do timed mode.

Every question has a suggested solutions videos after you complete the question. This video explains to you the steps to take to answer the question and provides tips and tricks.

Once you're done with the practice questions, move on to the next checkpoint.

Now, let’s get started on your practice questions.