Investigate properties of geometric figures

Ontario Grade 10 Mathematics

Analytic Geometry

Use side lengths, angles, and diagrams to recognize and solve problems about triangles and quadrilaterals.

A shape is more than its outline. Its side lengths, angle measures, and parallel sides give clues about what kind of figure it is and what must be true about it. In this lesson, you will review angle facts, use them to investigate triangles and quadrilaterals, and explain conclusions with clear reasoning. A drawing can help you notice a pattern, but a drawing alone does not prove a measurement. Look for given information and properties that apply to the figure.

What you will learn

Describe and compare properties of triangles and quadrilaterals.
Use angle relationships to find unknown angles in a figure.
Support a conclusion by connecting it to a known property.

1. Review: sides, angles, and parallel lines

A line segment is a straight part of a line with two endpoints. The length of a segment tells how long it is. An angle is formed when two segments or lines meet. Its measure tells how wide the opening is, in degrees.

Parallel lines are lines in the same plane that never meet. A transversal is a line that crosses two other lines. When the crossed lines are parallel, matching angles made by the transversal are equal. Angles inside the parallel lines on the same side of the transversal add to a straight angle.

A straight angle measures

180^\circ

. Angles around a point add to

360^\circ

. These facts help us find missing angles. For example, if two angles form a straight angle and one measures

65^\circ

, the other is

115^\circ

because their measures add to

180^\circ

.

When investigating a figure, separate what is given from what you can conclude. A sketch may look like it has equal sides, but unless that is marked, measured, or established by a property, do not assume the sides are equal.

\text{angles on a straight line}=180^\circ

A straight angle measures $180^\circ$ ; a full turn around a point measures $360^\circ$ .
Markings on a diagram show information such as equal sides, equal angles, or parallel sides.
Use a property or given measurement, not just the appearance of a sketch, to justify a conclusion.

2. Investigate triangles and quadrilaterals

A polygon is a closed, flat figure made from straight sides. A triangle has three sides. A quadrilateral has four sides. Polygons can be sorted by their properties, such as the number of sides, equal side lengths, equal angles, or parallel sides.

The interior angles are the angles inside a polygon. The three interior angles of any triangle add to

180^\circ

. This lets you find a missing angle when the other two are known. For instance, if a triangle has angles of

48^\circ

and

72^\circ

, its third angle is

60^\circ

.

A quadrilateral’s interior angles add to

360^\circ

. One way to understand this is to draw a diagonal from one vertex to the opposite vertex. The diagonal divides the quadrilateral into two triangles. Each triangle has an angle sum of

180^\circ

, so together the angles total

360^\circ

.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Opposite angles in a parallelogram are equal. Angles next to each other add to

180^\circ

. Its opposite sides are also equal in length. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides. A square has four right angles and four equal sides, so it has properties of both a rectangle and a rhombus.

These names describe properties, not just how a figure is turned or positioned. A tilted rectangle is still a rectangle if it has four right angles and opposite sides parallel. Use the defining properties to identify a shape, even when its drawing looks unfamiliar.

\text{quadrilateral interior-angle sum}=360^\circ

Triangle interior angles add to $180^\circ$ .
Quadrilateral interior angles add to $360^\circ$ .
A parallelogram has two pairs of parallel opposite sides, equal opposite sides, and equal opposite angles.
A square is both a rectangle and a rhombus.

3. Use properties to make and explain conclusions

A useful investigation follows a simple pattern: identify the figure, list the information, choose a property that applies, and use it to reach a conclusion. If a figure is marked as a parallelogram, you may use parallelogram properties. If it is only a four-sided sketch, you cannot assume it is a parallelogram.

For an unknown angle, write down the angle relationship before calculating. If the unknown angle and a known angle are adjacent angles in a parallelogram, they add to

180^\circ

. If they are opposite angles, they are equal. Choosing the correct relationship matters as much as doing the arithmetic correctly.

You can also investigate by comparing several figures. Record what stays the same and what changes. For example, rectangles can have different side lengths, but each has four right angles. This comparison helps distinguish a necessary property, which every figure of a type has, from a feature that may vary.

Name the figure and identify the information before choosing a rule.
Explain why a property applies to the specific figure.
Do not confuse a feature that every figure has with one that only some examples happen to show.

4. Try it yourself

For each question, show which property you use. A short sentence with a calculation is enough. If the diagram is not drawn to scale, trust the labels and given facts instead of estimating from its appearance.

Practice: A triangle has angles of

39^\circ

and

86^\circ

. Find its third angle. A quadrilateral has three interior angles measuring

91^\circ

,

104^\circ

, and

78^\circ

. Find its fourth angle. Then explain why the angle sum you used applies.

Practice: A parallelogram has one angle measuring

67^\circ

. State the measure of its opposite angle and of an angle next to it. Identify the property used for each answer.

Write the relevant angle sum or parallelogram property first.
Check that your answer fits the total angle measure or relationship.

Properties to use when investigating figures

Figure	Useful properties
Triangle	Three sides; interior angles add to $180^\circ$ .
Quadrilateral	Four sides; interior angles add to $360^\circ$ .
Parallelogram	Opposite sides are parallel and equal; opposite angles are equal; adjacent angles add to $180^\circ$ .
Rectangle	A parallelogram with four right angles.
Rhombus	A parallelogram with four equal sides.
Square	Four equal sides and four right angles.

Worked example

Find the angles in a parallelogram

In parallelogram

ABCD

, angle

A

measures

68^\circ

. Find angles

B

,

C

, and

D

.

Choose the properties
In a parallelogram, opposite angles are equal. Angles next to each other add to $180^\circ$ . Since $A$ and $C$ are opposite, they have the same measure. Angles $A$ and $B$ are next to each other, so their measures add to $180^\circ$ .
$\angle C=\angle A,\qquad \angle A+\angle B=180^\circ$
Find angle B
Subtract the known angle from $180^\circ$ to find the angle next to it. This uses the fact that the two adjacent angles add to a straight angle.
$\angle B=180^\circ-68^\circ=112^\circ$
Find angles C and D
Angle $C$ is opposite angle $A$ , so it measures $68^\circ$ . Angle $D$ is opposite angle $B$ , so it measures $112^\circ$ .
$\angle C=68^\circ,\qquad \angle D=112^\circ$

Answer:

\angle B=112^\circ

,

\angle C=68^\circ

, and

\angle D=112^\circ

.

Check: The four angles total

68^\circ+112^\circ+68^\circ+112^\circ=360^\circ

, as the interior angles of a quadrilateral should.

Common mistakes and how to avoid them

Assuming a shape is a rectangle because it looks like one in the drawing.

Correction: Check the information or markings. A rectangle must have four right angles.

Using

180^\circ

as the angle sum for a quadrilateral.

Correction: A quadrilateral has four interior angles that add to

360^\circ

. The

180^\circ

sum applies to a triangle.

Assuming all four angles in a parallelogram are equal.

Correction: Only opposite angles are equal in every parallelogram. Adjacent angles add to

180^\circ

.

Giving a number without saying why the rule applies.

Correction: Name the figure and state the property that connects the given information to your answer.

Lesson summary

Use side lengths, angle measures, and parallel sides to identify and compare figures.
Triangle angles add to $180^\circ$ ; quadrilateral angles add to $360^\circ$ .
In a parallelogram, opposite angles are equal and adjacent angles add to $180^\circ$ .
A conclusion is stronger when you name the property that supports it.

Check your understanding

Question 1

A triangle has angles of

53^\circ

and

64^\circ

. What is its third angle?

$53^\circ$
$63^\circ$
$117^\circ$
$243^\circ$

Show answer and explanation

63^\circ

The angles in a triangle add to

180^\circ

. Subtract the known angles:

180^\circ-53^\circ-64^\circ=63^\circ

.

Question 2

A parallelogram has an angle of

121^\circ

. What is the measure of an angle next to it?

$59^\circ$
$121^\circ$
$180^\circ$
$239^\circ$

Show answer and explanation

59^\circ

Adjacent angles in a parallelogram add to

180^\circ

, so the neighbouring angle is

180^\circ-121^\circ=59^\circ

.

Question 3

Which statement must be true for every square?

It has exactly one pair of parallel sides.
It has four equal sides and four right angles.
Its adjacent angles are unequal.
It has three sides.

Show answer and explanation

It has four equal sides and four right angles.

A square has four equal sides and four right angles. It also has two pairs of parallel opposite sides.

Key terms

Polygon: A closed, flat figure made from straight sides.
Interior angle: An angle inside a polygon, formed where two sides meet.
Parallel lines: Lines in the same plane that never meet.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Right angle: An angle that measures $90^\circ$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G11. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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