Solve problems using properties of angles and triangles

Alberta Grade 11 Mathematics

Geometry

Choose a property, form an equation, and check the result

Angles and triangles appear in signs, plans, and building designs. A diagram may show only some of the angle measures. Properties that are always true can help you find missing measures. First identify the relationship, then write an equation, solve it, and check your answer. A sketch may not be drawn to scale, so rely on the information given rather than estimating from its appearance.

What you will learn

Identify angle relationships shown or stated in a problem.
Use angle properties to write and solve equations.
Use triangle angle sums and isosceles-triangle properties.
Check that an answer fits the given relationships.

Review: angles, variables, and equations

An angle measures the amount of turn between two rays that share an endpoint. The endpoint is called the vertex. Angle measures are usually given in degrees, written with the symbol

^\circ

. A straight angle measures

180^\circ

. A full turn around a point measures

360^\circ

.

A variable is a letter that represents an unknown number. An equation states that two quantities are equal. For example, if two angles form a straight angle and one measures

65^\circ

, the other angle, called

x

, satisfies

x+65=180

. Subtracting 65 from both sides gives

x=115

.

Use the diagram and the words in the problem to decide which relationship applies. A drawing can help you see a relationship, but the stated conditions and angle properties justify the calculation.

180^\circ;\quad 360^\circ

Angle measures are written in degrees.
Use a variable to represent an unknown measure.
An equation should represent a property that applies to the diagram.

Angle relationships

Adjacent angles share a vertex and one side. If their other sides point in opposite directions and form a straight line, the angles are supplementary. Supplementary angles have measures that add to

180^\circ

.

Angles around a point make a full turn, so their measures add to

360^\circ

. Use this property when the angles at one point cover the complete turn.

When two straight lines cross, the opposite angles are called vertically opposite angles. Vertically opposite angles have equal measures. Each pair of adjacent angles forms a straight line, so the pair adds to

180^\circ

.

Parallel lines never meet. A transversal is a line that crosses two or more other lines. If a transversal crosses two parallel lines, corresponding angles are equal. Corresponding angles occupy matching positions at the two crossings. Interior angles on the same side of the transversal add to

180^\circ

. Use these properties only when the lines are stated or marked as parallel.

For example, if vertically opposite angles are represented by

3x+5

and

80

, their measures are equal. The equation is

3x+5=80

. Solve for

x

, then substitute its value into the original expression to check the angle measure.

a+b=180^\circ;\quad a+b+c=360^\circ

Angles on a straight line add to $180^\circ$ .
Angles around a point add to $360^\circ$ .
Vertically opposite angles are equal.
For parallel lines, corresponding angles are equal, and same-side interior angles add to $180^\circ$ .

Triangle properties

A triangle is a closed figure with three sides and three interior angles. Its interior angles always add to

180^\circ

. If two angles are known, subtract their sum from

180^\circ

to find the third.

An isosceles triangle has two equal sides. The angles opposite those sides are equal. The equal sides meet at one vertex; the two remaining angles are the equal angles. This property can be combined with the triangle angle sum.

An exterior angle is an angle outside a triangle formed by extending one of its sides. Its measure equals the sum of the two interior angles that are not beside it. It is also supplementary to the interior angle beside it.

When solving a triangle problem, mark any stated equal sides or angles. Choose the property that follows from the markings or information. An angle in a triangle must be greater than

0^\circ

and less than

180^\circ

.

A+B+C=180^\circ

The interior angles of a triangle total $180^\circ$ .
In an isosceles triangle, the angles opposite the equal sides are equal.
A triangle's exterior angle equals the sum of its two non-adjacent interior angles.

Plan a solution and check it

A reliable solution identifies the relationship, calculates the unknown, and checks the result. State why the relationship applies. For example, two angles are equal because they are vertically opposite, or three angles total

180^\circ

because they are the interior angles of a triangle.

When an angle is written as an expression, solve the equation and substitute the value back into that expression. Check that the measures satisfy the relationship and that each angle is a sensible positive measure.

Try this: two adjacent angles form a straight line. Their measures are

4x^\circ

and

2x^\circ

. Since their sum is

180^\circ

, write

4x+2x=180

. Thus

x=30

, and the angles measure

120^\circ

and

60^\circ

. Their sum is

180^\circ

, so the result checks.

Choose a property before writing an equation.
Substitute your answer to check the original relationship.
Check that angle measures are positive and fit the stated conditions.

Worked example

Find the angles at a crossing

Two straight lines cross. One of the angles measures

68^\circ

. Find the vertically opposite angle and each adjacent angle.

Two crossing lines

One angle is 68°. The opposite angle has the same measure, and each adjacent angle is supplementary to it.

Use the opposite-angle property
The vertically opposite angle equals the given angle because opposite angles formed by two crossing straight lines are equal.
$68^\circ$
Find the adjacent angles
An adjacent angle and the given angle form a straight line, so their measures add to $180^\circ$ . The two adjacent angles are also vertically opposite each other, so they have the same measure.
$180^\circ-68^\circ=112^\circ$

Answer: The vertically opposite angle is

68^\circ

. Each adjacent angle is

112^\circ

.

Check: The opposite angles are equal. A given angle and either adjacent angle add to

68^\circ+112^\circ=180^\circ

, as required.

Worked example

Solve for an unknown with parallel lines

A transversal crosses two parallel lines. A pair of corresponding angles measures

(3x+12)^\circ

and

75^\circ

. Find

x

and the angle measure.

Parallel lines and a transversal

The indicated corresponding angles are equal because the two lines are parallel. Each measures 75°.

Set the corresponding angles equal
The lines are stated to be parallel, so corresponding angles have equal measures. Set the two expressions equal.
$(3x+12)^\circ=75^\circ$
Solve for the variable
Subtract 12 from both sides, then divide both sides by 3. These operations keep the equation balanced.
$3x=63,\quad x=21$
Substitute to find the angle
Substitute 21 for $x$ in the expression. The result matches the other corresponding angle.
$3(21)+12=75$

Answer:

x=21

. Each corresponding angle measures

75^\circ

.

Check: Substituting

x=21

gives

3(21)+12=75

. The matching measures agree with the corresponding-angle property.

Worked example

Use the properties of an isosceles triangle

An isosceles triangle has two equal sides. The angle between those sides measures

44^\circ

. Find each of the other two angles.

Angles in an isosceles triangle

The equal sides meet at the 44° vertex angle. The other two angles are equal.

Represent the equal angles
The two equal sides meet at the $44^\circ$ angle. The two remaining angles are opposite the equal sides, so they are equal. Let each one be $x$ degrees.
$x+x+44=180$
Use the triangle angle sum
The interior angles of a triangle add to $180^\circ$ . Subtract 44 from 180, then share the remaining measure equally between the two equal angles.
$2x=136,\quad x=68$

Answer: Each of the other two angles measures

68^\circ

.

Check: The angles add to

68^\circ+68^\circ+44^\circ=180^\circ

. The two angles opposite the equal sides are equal.

Common mistakes and how to avoid them

Assuming two angles are equal because they look equal in a sketch.

Correction: Use equality only when it follows from information in the problem or a stated angle property.

Using

180^\circ

for all the angles around a point.

Correction: Angles on a straight line add to

180^\circ

. Angles around a point make a full turn and add to

360^\circ

.

Treating every pair of angles made by a transversal as equal.

Correction: For the parallel-line angle properties, first confirm that the lines are stated or marked to be parallel.

Using the equal-angle property for a triangle whose sides are not stated to be equal.

Correction: Use equal opposite angles only when the triangle is stated to be isosceles or its equal sides are marked.

Lesson summary

Angles on a straight line total $180^\circ$ , and angles around a point total $360^\circ$ .
Vertically opposite angles are equal.
For parallel lines, corresponding angles are equal and same-side interior angles total $180^\circ$ .
The interior angles of a triangle total $180^\circ$ . In an isosceles triangle, the angles opposite the equal sides are equal.
Explain which property you used, then check that the measures satisfy it.

Check your understanding

Question 1

Two angles on a straight line measure

5x^\circ

and

4x^\circ

. What is the smaller angle?

$20^\circ$
$80^\circ$
$100^\circ$
$180^\circ$

Show answer and explanation

80^\circ

The angles add to

180^\circ

, so

5x+4x=180

. Thus

x=20

, and the smaller angle is

4x=80^\circ

.

Question 2

A triangle has two angles measuring

53^\circ

and

71^\circ

. What is its third angle?

$56^\circ$
$66^\circ$
$124^\circ$
$256^\circ$

Show answer and explanation

56^\circ

The third angle is

180^\circ-(53^\circ+71^\circ)=56^\circ

.

Question 3

An isosceles triangle has two equal sides. One angle opposite an equal side measures

42^\circ

. What is the other angle opposite an equal side?

$42^\circ$
$48^\circ$
$96^\circ$
$138^\circ$

Show answer and explanation

42^\circ

The angles opposite the equal sides of an isosceles triangle are equal. The other angle is also

42^\circ

.

Key terms

Adjacent angles: Two angles that share a vertex and one side.
Supplementary angles: Two angles whose measures add to $180^\circ$ .
Vertically opposite angles: Opposite angles formed when two straight lines cross. They have equal measures.
Parallel lines: Lines in the same plane that never meet.
Transversal: A line that crosses two or more other lines.
Isosceles triangle: A triangle with two equal sides.
Exterior angle: An angle outside a triangle formed by extending one of its sides.

Continue through Math 20-2

View the complete Math 20-2 Alberta Grade 11 Mathematics curriculum and lessons

Tutor support for G2 · Solve problems using properties of angles and triangles

Working on G2 · Solve problems using properties of angles and triangles in Math 20-2? Bring the question and your own attempt. In a live session, you can show each step of your attempt and compare it with a worked example. Ask about the first step that does not make sense, then try a similar question yourself.

Explore math tutoring · Request an eligible free 30-minute introduction. Your request is matched by subject and language; a tutor must accept before a time is confirmed.

برای مبحث G2 · مسئله‌ها را با ویژگی‌های زاویه و مثلث حل کنید در درس Math 20-2، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید گام‌های تلاش خود را نشان دهید و آن را با یک نمونهٔ حل‌شده مقایسه کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation G2. 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.

Official curriculum reference

Report a correction or ask a question