Math resources Geometry Angles

Straight angle

# Straight angle

Here we will learn about straight angles, including the sum of straight angles, how to find missing angles, and using these angle facts to generate equations and solve problems.

Students will first learn about straight angles as part of geometry in 7 th grade.

## What are straight angles?

A straight angle is an angle on a straight line. The measure of a straight angle is exactly 180^{\circ}. It is also called a flat angle.

When two rays or line segments extend from a common endpoint in opposite directions, they create a straight angle. The endpoint forms the vertex of the angle.

A straight angle can also refer to the combined measure of angles arranged in a way that they form a straight line and collectively add up to 180^{\circ}.

For example, let’s take the three angles of a, b, and c.

If we move these three angles so that each vertex meets, we get an arrangement that looks like this:

These three angles create a straight line.

By adding together a=90^{\circ}, b=38^{\circ} and c=52^{\circ}, we can see the angle measurements add up to 180^{\circ}. Therefore, they create a straight angle.

## Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – Geometry (7.G.B.5)
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

## How to find missing angle measurements on a straight angle

In order to find missing angle measurements on a straight angle:

1. Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.
2. Simplify by collecting like terms.
3. Solve the equation.

## Straight angle examples

### Example 1: obtuse angle

The angles below create a linear pair, which is a pair of angles that form a straight line. Calculate the missing angle x.

1. Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

x+78=180

2Simplify by collecting like terms.

Here, there are no terms to collect without solving the equation.

3Solve the equation.

### Example 2: acute angle

AB is a straight line through O. Calculate the missing angle x.

Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

Simplify by collecting like terms.

Solve the equation.

### Example 3: right angle

AB is a straight line through O. Calculate the missing angle x.

Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

Simplify by collecting like terms.

Solve the equation.

### Example 4: vertically opposite angles

AB and CD are straight lines. By calculating the value of y, determine the value of x.

Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

Simplify by collecting like terms.

Solve the equation.

### Example 5: forming and solving equations

AB is a straight line through O. Calculate the size of all the angles that make up the line AB.

Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

Simplify by collecting like terms.

Solve the equation.

### Example 6: circles and tangents

AB is a tangent to the circle with center C. The tangent intersects the circle at the point O on the circumference. Use this information to calculate the value of x.

Form an equation using the rule: The sum of angles in a straight angle =\bf{180^{\circ}}.

Simplify by collecting like terms.

Solve the equation.

### Teaching tips for straight angles

• Show real life examples of straight angles, like the edges of a book or the arms of a clock at 6 o’clock.

• Build upon students’ prior knowledge of angles, such as right angles and acute angles, to introduce straight angles. Highlight how straight angles relate to different types of angles they have previously learned about.

• Use formative assessment strategies such as exit tickets, multiple choice quizzes, or mini whiteboard activities to gauge student understanding of straight angles. Provide immediate feedback on worksheets during class to address misconceptions and reinforce key concepts.

### Easy mistakes to make

• Thinking that the sum of straight angles is equal to \bf{360^{\circ}}
Some students may incorrectly remember straight angles as 360^{\circ}, rather than 180^{\circ}. The sum of straight angles is half of a revolution, which is 180^{\circ}.

• Using a protractor
When you are asked to calculate a missing angle, a common error is to use a protractor to measure the angle. When using angle facts to determine angles, diagrams are deliberately not drawn to scale unless it is a 90 degree angle or a 180 degree angle as these are important angles to recognize. You should not use a protractor for this style of question.

• Confusing straight angles with reflex angles
Students might confuse straight angles with reflex angles, which are angles greater than 180^{\circ} and less than 360^{\circ}.

### Practice straight angles questions

1. The two angles shown below form a linear pair. Calculate the size of angle x.

95^{\circ}

25^{\circ}

295^{\circ}

115^{\circ}
\begin{aligned}x+65&=180 \\\\ x&=180-65 \\\\ x&=115^{\circ} \end{aligned}

2. AOB is a straight line. Calculate the size of angle x.

238^{\circ}

32^{\circ}

58^{\circ}

122^{\circ}
\begin{aligned}122+x&=180 \\\\ x&=180-122 \\\\ x&=58^{\circ} \end{aligned}

3. Calculate the size of the angle 2x. Hence find the value of x.

52^{\circ}

26^{\circ}

76^{\circ}

116^{\circ}
\begin{aligned}2x+90+38&=180 \\\\ 2x+128&=180 \\\\ 2x&=180-128 \\\\ 2x&=52 \\\\ x&=52\div{2} \\\\ x&=26^{\circ} \end{aligned}

4. AB and CD are straight lines. Calculate the size of angle BOD. Hence find the value of x.

40^{\circ}

220^{\circ}

12^{\circ}

8^{\circ}
\begin{aligned}5x+140&=180 \\\\ 5x&=180-140 \\\\ 5x&=40 \\\\ x&=40\div{5} \\\\\ x&=8^{\circ} \end{aligned}

5. AOB is a straight line. By finding the value for x, calculate the size of each angle in the diagram below.

x=12.5^{\circ}, COD=72.5^{\circ}, DOE=7.5^{\circ}, EOB=75^{\circ}

x=11.1^{\circ}, COD=55.5^{\circ}, DOE=8.9^{\circ}, EOB=69.4^{\circ}

x=35^{\circ}, COD=175^{\circ}, DOE=15^{\circ}, EOB=165^{\circ}

x=2.05^{\circ}, COD=10.2^{\circ}, DOE=17.95^{\circ}, EOB=33.18^{\circ}
35+5x+20-x+4x+25=180

\begin{aligned}8x+80&=180 \\\\ 8x&=180-80 \\\\ 8x&=100 \\\\ x&=100\div{8} \\\\ x&=12.5^{\circ} \end{aligned}

5x=5\times{12.5}=62.5^{\circ}

20-x=20-12.5=7.5^{\circ}

4x+25=4\times{12.5}+25=75^{\circ}

6. The circle with center C has a tangent at point O. Calculate the value of x correct to the nearest hundredth ( 2 decimal places).

19.11^{\circ}

9.11^{\circ}

10.59^{\circ}

5.29^{\circ}
\begin{aligned}9x+8+90&=180 \\\\ 9x+98&=180 \\\\ 9x&=180-98 \\\\ 9x&=82 \\\\ x&=82\div{9} \\\\ x&=9.\dot{1}=9.11^{\circ}\text{ (2 dp)} \end{aligned}

## Straight angle FAQs

What is a straight angle?

A straight angle is an angle on a straight line whose measure is exactly 180^{\circ}. It is also called a flat angle.

What is the difference between a straight angle and supplementary angles?

A straight angle is a specific type of angle that measures 180^{\circ} (or radians) and forms a straight line. Supplementary angles are angle pairs whose measures add up to 180^{\circ} but they don’t have to form a straight line. They can be adjacent or non-adjacent angles.

What is the difference between a straight angle and a full angle?

A straight angle measures 180^{\circ} while a full angle, or a complete angle, measures 360^{\circ}. Two straight angles equal one full angle.

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