# 5.1: Right, Acute and Obtuse Angles

• If (|measuredangle AOB| = dfrac{pi}{2}), we say that (angle AOB) is right;
• If (|measuredangle AOB| < dfrac{pi}{2}), we say that (angle AOB) is acute;
• If (|measuredangle AOB| > dfrac{pi}{2}), we say that (angle AOB) is obtuse.

On the diagrams, the right angles will be marked with a little square, as shown.

If (angle AOB) is right, we say also that ([OA)) is perpendicular to ([OB)); it will be written as ([OA) perp [OB)). From Theorem 2.4.1, it follows that two lines ((OA)) and ((OB)) are appropriately called perpendicular, if ([OA) perp [OB)). In this case we also write ((OA) perp (OB)).

Exercise (PageIndex{1})

Assume point (O) lies between (A) and (B) and (X e O). Show that (angle XOA) is acute if and only if (angle XOB) is obtuse.

Hint

By Axiom IIIb and Theorem 2.4.1, we have (measuredangle XOA - measuredangle XOB equiv pi). Since (|measuredangle XOA|, |measuredangle XOB| le pi), we get that (|measuredangle XOA| + |measuredangle XOB| = pi). Hence the statement follows.

## Identifying Acute, Right, and Obtuse Angles Worksheets

Make headway with our acute, right, and obtuse angles worksheets that help familiarize children with the three types of angles with a variety of exercises like determining the types of angles visually, counting the angles of each type, matching angles to the measures and figures, identifying angles formed by the hands of a clock, comprehending angles with real-life objects, identifying angles in multiple rays and sorting them, and revision worksheets to revisit and test your 4th grade and 5th grade students. Try our free worksheets to get the ball rolling.

Give children a head-start with this acute, right, and obtuse angle chart showcasing the three types of angles. Watch the position of the ray the angle is acute if < 90°, right if it is 90° and obtuse if > 90°.

Observe, identify, and label the angles formed by the rays in the first part, and in the shapes in the latter part of this set of printable worksheets on identifying angles as acute, obtuse, and right.

How many angles of each type are there in the polygons featured in these pdf counting angles worksheets? Children are expected to identify the type of angles, and count the number of acute, obtuse, and right angles they find in each shape.

Grade 4 and Grade 5 kids measure the angles, identify their type, and make one-to-one correspondence between the type of angles, their measures and figures in this batch of worksheets on matching angles.

Add an element of fun by opting for these types of angles in a clock worksheets. Get students to draw the minute hand according to the specified time, and determine the type of angle formed between the minute and hour hand of each clock.

What is the type of angle formed by the roof of a house or a slice of pizza? Classify everyday objects based on the indicated angle as acute, obtuse, and right angles with this collection of activity worksheet pdfs.

Work your way through these printable sorting angles in multiple rays worksheets by identifying the angles formed. Sort them based on their type as acute, obtuse, and right and write them in the appropriate columns.

Review concepts with this compilation of exercises that make a compulsive print for your 4th grade and 5th grade students to become adept in recognizing and distinguishing acute, obtuse, and right angles.

## Difference Between Acute Angle and Obtuse Angle

Angles are defined as the shape formed by the intersection of two straight lines. The straight line segments are called the sides, and the point of intersection is known as the vertex. The size of an angle is measured by the separation of its sides around the vertex. The measure of angle can also be defined mathematically as the ratio between the arc created by the angle and the radius of the arc.

Radians are the standard unit of measurement of angles, though degrees and grad are also used. Angles are typically used as a measurement of rotation or angular separation.

Angles are an important concept in the study of geometry, and they are classified based on their special features. An angle is acute if its magnitude is less than π/2 rad or 90° (i.e. 0≤θ≤π/2 rad). An angle is called an obtuse angle if its magnitude is in between π/2 rad or 90° and π rad or 180°.

Acute Angle Obtuse Angle

The other side of the obtuse angles and acute angles always create reflex angles.

What is the difference between Obtuse Angle and Acute Angle?

• Acute angle is an angle with size less than π/2 rad or 90°

• Obtuse angle is an angle with size between π/2 rad or 90° and π rad or 180°

• In other words, an angle between a straight angle and a right angle is known as an obtuse angle, and an angle less than a right angle is known as an acute angle.

## 5.1: Right, Acute and Obtuse Angles

Input 3 triangle side lengths (A, B and C), then click "ENTER". This calculator will determine whether those 3 sides will form an equilateral, isoceles, acute, right or obtuse triangle or no triangle at all.

Without Using The Calculator
When given 3 triangle sides, to determine if the triangle is acute, right or obtuse:

2) Sum the squares of the 2 shortest sides.

3) Compare this sum to the square of the 3rd side.

if sum > 3rd side²   Acute Triangle

if sum = 3rd side²   Right Triangle

sum of the squares of the short sides = 25 + 36 = 61

3, 4 and 5 Squaring each side = 9, 16 and 25

sum of the squares of the short sides = 9 + 16 = 25

3, 7 and 9 Squaring each side = 9, 49 and 81

sum of the squares of the short sides = 9 + 49 = 58

58 For determining if the 3 sides can even form a triangle, the triangle inequality theorem states that the longest side must be shorter than the sum of the other 2 sides.

## 5.1: Right, Acute and Obtuse Angles

Because the angles of a triangle add up to 180°, at least two of them must be acute (less than 90°). In an acute triangle all angles are acute. A right triangle has one right angle, and an obtuse triangle has one obtuse angle.

The altitude corresponding to a side is the perpendicular dropped to the line containing that side from the opposite vertex. The bisector of a vertex is the line that divides the angle at that vertex into two equal parts. The median is the segment joining a vertex to the midpoint of the opposite side. See Figure 1.

Figure 1: Notations for an arbitrary triangle of sides a, b, c and vertices A, B, C. The altitude corresponding to C is , the median is , the bisector is . The radius of the circumscribed circle is R, that of the inscribed circle is r.

Every triangle also has an inscribed circle tangent to its sides and interior to the triangle (in other words, any three nonconcurrent lines determine a circle). The center of this circle is the point of intersection of the bisectors. We denote the radius of the inscribed circle by r.

Every triangle has a circumscribed circle going through its vertices in other words, any three noncollinear points determine a circle. The point of intersection of the medians is the center of mass of the triangle (considered as an area in the plane). We denote the radius of the circumscribed circle by R.

Introduce the following notations for an arbitrary triangle of vertices A, B, C and sides a, b, c (see Figure 1). Let , and be the lengths of the altitude, bisector and median originating in vertex C, let r and R be as usual the radii of the inscribed and circumscribed circles, and let s=½(a+b+c). Then:

A triangle is equilateral if all its sides have the same length, or, equivalently, if all its angles are the same (and equal to 60°). It is isosceles if two sides are the same, or, equivalently, if two angles are the same. Otherwise it is scalene.

For an equilateral triangle of side a we have:

area=¼a ,
r= a ,
R= a ,
ha ,

where h is any altitude. The altitude, the bisector and the median for each vertex coincide.

For an isosceles triangle, the altitude for the unequal side is also the corresponding bisector and median, but this is not true for the other two altitudes. Many formulas for an isosceles triangle of sides a, a, c can be immediately derived from those for a right triangle of legs a, ½c (see Figure 2, left).

Figure 2: Left: An isosceles triangle can be divided into two congruent right triangles. Right: notations for a right triangle.

For a right triangle the hypothenuse is the longest side opposite the right angle the legs are the two shorter sides, adjacent to the right angle. The altitude for each leg equals the other leg. In Figure 2 (right), h denotes the altitude for the hypothenuse, while m and n denote the segments into which this altitude divides the hypothenuse.

The following formulas apply for a right triangle:

A+B=90°
r=ab/(a+b+c)
a=c sin A = c cos B
mc=b
area=½ab
c =a +b (Pythagoras)
Rc
b=c sin B = c cos A
nc=a
hc=ab

The hypothenuse is a diameter of the circumscribed circle. The median joining the midpoint of the hypothenuse (the center of the circumscribed circle) to the right angle makes angles 2A and 2B with the hypothenuse.

• In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. This follows from the law of sines.
• (Ceva's Theorem: see Figure 3, left.) In a triangle ABC, let D, E and F be points on the lines BC, CA and AB, respectively. Then the lines AD, BE and CF are concurrent if and only if the signed distances BD, CE, . satisfy

This is so in three important particular cases: when the three lines are the medians, when they are the bisectors, and when they are the altitudes.

Figure 3: Left: Ceva's Theorem. Right: Menelaus's Theorem.

Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.

Question 1.
Find the complement of each of the following angles :
i) The complement of 70°
Solution:

The complement of the angle 20° = 90° – 20° = 70°

ii)
The complement of the angle 63° = 90° – 63° = 27°

iii)
The complement of the angle 57° = 90° – 57° = 33°

Question 2.
Find the supplement of each of the following angles :
Solution:
i)
The supplement of the angle 105°= 180° – 105° = 75°

ii)
The supplement of the angle 87°= 180° – 87° = 93°

iii) <
The supplement of the angle 154°= 180° – 154° = 26°

Question 3.
Identify which of the following pairs of angles are complementary and which are supplementary.
i) 65°, 115°
65° + 115° = 180°
∴ This pair is supplementary angles.

ii) 63°, 27°
63°+ 27° = 90°
∴ This pair is complementary angles.

iii) 112°, 68°
112°+68°= 1800
∴ This pair is supplementary angles.

iv) 130°, 50°
130° + 50° = 180°
∴ This pair is supplementary angles.

v) 45°, 45°
45° + 45° = 90°
∴ This pair is complementary angles.

vi) 80°, 10°
80°+ 10° = 90°
This pair is complementary angles.

Question 4.
Find the angle which is equal to its complement.
Solution:
Let one of the complement angle be x°
Its complement be = 90° – x
∴ According to the question
x° = 90° – x°
x° + x° = 90°
2x° = 90°
x = 90/2 = 45°

Question 5.
Find the angle which is equal to its supplement.
Solution:
Let one of supplement be x°
Another supplement be = 180° – x°
According to the question = x° = 180° – x°
x° + x° = 180°
2x° = 180°
x° = 180/2 = 90°

Question 6.
In the given figure, ∠1 and ∠2 are supplementary angles.
Solution:
If ∠1 is decreased, what changes should take place in ∠2 so that both the angles still remain supplementary?

If ∠1 is decreased the ∠2 will be increased.

Question 7.
Can two angles be supplementary if both of them are :
i) acute?
No, two acute angles cannot be supplementary. [∵ acute angles is ∠90°]

ii) obtuse?
No, two obtuse angles cannot be supplementary. [∵ obtuse angles is ∠90°].

iii) right?
Yes, Two right angles always supplementary. [∵ right angles is = 90°].

Question 8.
An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°?
Solution:
If an angle is greater than 45° then its complement should be less than 45°.

Question 9.

i) Is ∠1 adjacent to ∠2?
Yes, ∠1 is adjacent to ∠2
ii) Is ∠AOC adjacent to ∠AOE?
No, ∠AOC is not adjacent to ∠AOE.
iii) Do ∠COE and ∠EOD form a linear pair?
Yes, ∠COE and ∠EOD are linear pairs.
iv) are ∠BOD and ∠DO A supplementary?
Yes, ∠BOD and ∠DOA are supplementary.
v) Is ∠1 vertically opposite to ∠4?
Yes, ∠1 is vertically opposite to ∠4.
vi) What is the vertically opposite angle of ∠5?
The vertically opposite angle of ∠5 is ∠2 + ∠3 ie., ∠COB.
∠COE + ∠EOB = ∠COB

Question 10.
Indicate which pairs of angles are :

i) Vertically opposite angles.
Vertically opposite angles are
∠1 and ∠4
∠5 and ∠2 + ∠3
ii) Linear pairs
Linear pairs
∠5 and ∠1
∠4 and ∠5

Question 11.
In the following figure, is ∠1 adjacent to ∠2 ? Give reasons.
Solution:

∠1 is not adjacent to ∠2 because their vertex is not common.

Question 12.
Find the values of the angles x, y, and z in each of the following :
Solution:
Given < = 55°

∠x = 55° (∵ vertically opposite angles)
∠y = 180° – 55° = 125° (∵ linear pair)
∠z = ∠y = 125°(∵ vertically opposite angles)

ii)
Given
∠AOC = 40°
∠EOB = 25° .
AOB is a straight angIe (∵ AOB is st line)
∠AOB = ∠AOC + ∠EOB
180° = 40° + ∠COE + 25°
180° = 65°+ ∠COE
∴ ∠COE = 180° – 65° = 115°
∠y + ∠z = 180°
∠z = 40° (∵ vertically opposite angles)
∴ y = 180° – 40° = 140°

Question 13.
Fill in the blanks :

1. If two angles are complementary, then the sum of their measures is 90°
2. If two angles are supplementary, then the sum of their measures is 180°
3. Two angles forming a linear pair are supplementary
4. If two adjacent angles are supplementary, they form a linear pair
5. If two lines intersect at a point, then the vertically opposite angles are always equal
6. If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are obtuse angles.

Question 14.
In the adjoining figure, name the following pairs of angles.

Solution:
i) Obtuse vertically opposite angles
∠AOD and ∠BOC

∠BOA and ∠AOE

iii) Equal supplementary angles
∠BOE and ∠EOD

iv) Unequal supplementary angles
∠AOE and ∠EOC

v) Adjacent angles that do not form a linear pair
∠AOB and ∠AOE, ∠AOE and ∠EOD and ∠EOD and ∠COD

Question 1.
Find the complement of each of the following angles:

Solution:
Since, the sum of the measures of an angle and its complement is 90°, therefore,

1. The complement of an angle of measure 20° is the angle of (90° – 20°), f.e., 70°.
2. The complement of an angle of measure 63° is the angle of (90° – 63°), i.e., 27°.
3. The complement of an angle of measure 57° is the angle of (90° – 57°), i.e., 33°.

Question 2.
Find the supplement of each of the following angles:

Solution:

1. Supplement of the angle 105° = 180° – 105° = 75°
2. Supplement of the angle 87° = 180° – 87° = 93°
3. Supplement of the angle 154° = 180° – 154° = 26°

Question 3.
Identify which of the following pairs of angles are complementary and which are supplementary.

1. Since, 65°+ 115° = 180°
So, this pair of angles are supplementary.
2. Since, 63°+ 27° = 90°
So, this pair of angles are complementary.
3. Since, 112° + 68° = 180°
So, this pair of angles are supplementary.
4. Since, 130°+50° = 180°
So, this pair of angles are supplementary.
5. Since, 45°+ 45° = 90°
So, this pair of angles are complementary.
6. Since, 80°+ 10° = 90°
So, this pair of angles are complementary.

Question 4.
Find the angle which is equal to its complement.
Solution:
Let the measure of the angle be x°. Then, the measure of its complement is given to be x°.
Since, the sum of the measures of an angle and its complement is 90°, therefore,
x° + x° = 90°
⇒ 2x° = 90°
⇒ x° = 45°
Thus, the required angle is 45°.

Question 5.
Find the angle which is equal to its supplement.
Solution:
Let the measure of the angle be x°. Then,
a measure of its supplement = x°
Since, the sum of the measures of an angle and its supplement is 180°, therefore,
x° + x° = 180°
⇒ 2x° =180°
⇒ x° = 90°
Hence, the required angle is 90°.

Question 6.
In the given figure, ∠ 1 and ∠ 2 are supplementary angles.

If ∠1 is decreased, what changes should take place in ∠ 2 so that both the angles still remain supplementary?
Solution:
∠ 2 will increase as much as ∠ 1 decreases.

Question 7.
Can two angles be supplementary if both of them are:

1. No! two acute angles cannot be a supplement.
2. No! Two obtuse angles cannot be supplementary.
3. Yes! Two right angles are always supplementary.

Question 8.
An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°.
Solution:
Since the sum of the measure of ah angle and its complement is 90°.
∴ The complement of an angle of measures 45° + x°,
where x > 0 is the angle of [90° – (45° + x°)] = 90° – 45° – x°= 45° – x°.
Clearly, 45° + x° > 45° – x°
Hence, the complement of an angle > 45° is less than 45°.

Question 9.

1. Is ∠1 adjacent to ∠2 ?
2. Is ∠ AOC adjacent to ∠ AOE?
3. Do ∠ COE and ∠ EOD form a linear pair?
4. Are ∠ BOD and ∠ DOA supplementary?
5. Is ∠ 1 vertically opposite to ∠ 4?
6. What is the vertically opposite angle of ∠ 5?
1. Yes ! ∠ 1 is adjacent to ∠ 2.
2. No ! ∠ AOC is not adjacent to ∠ AOE.
3. Yes! ∠ COE and ∠ EOD form a linear pair.
4. Yes ! ∠ BOD and ∠ DOA are supplementary.
5. Yes ! ∠ 1 is vertically opposite to ∠ 4.
6. The vertically opposite angle of ∠ 5 is ∠ 2 + ∠ 3, i.e., ∠ COB.

Question 10.
Indicate which pairs of angles are:

1. The pair of vertically opposite angles are ∠1, ∠4 ∠5, ∠2 + ∠3.
2. The pair of linear angles are ∠1, ∠5 ∠4, ∠5.

Question 11.
In the following figure, is ∠ 1 adjacent to ∠ 2? Give reasons.

Solution:
∠1 is not adjacent to ∠2 because they have no common vertex.

Question 12.
Find the values of the angles x, y, and z in each of the following:

Solution:

Question 13.
Fill in the blanks:

1. If two angles are complementary, then the sum of their measures is
2. If two angles are supplementary, then the sum of their measures is
3. Two angles forming a linear pair are
4. If two adjacent angles are supplementary, they form a
5. If two lines intersect at a point, then the vertically opposite angles are always
6. If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are
1. 90°
2. 180°
3. supplementary
4. linear pair
5. equal
6. obtuse angles

Question 14.
In the adjoining figure, name the following pairs of angles.

1. Obtuse vertically opposite angles
3. Equal supplementary angles
4. Unequal supplementary angles
5. Adjacent angles that do not form a linear pair.
1. Obtuse vertically opposite angles are ∠AOD and ∠BOC.
2. Adjacent complementary angles are ∠BOA and ∠AOE.
3. Equal supplementary angles are ∠BOE and ∠EOD.
4. Unequal supplementary angles are ∠BOA and ∠AOD, ∠BOC and ∠COD, ∠EOA, and ∠EOC.
5. Adjacent angles that do not form a linear pair are ∠AOB and ∠AOE, ∠AOE and ∠EOD ∠EOD and ∠COD.

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The angle shown in the provided figure is an acute angle.

### Explanation of Solution

Given Information:

An angle as shown in the below figure:

The angles can be classified on the basis of the degree measure.

A right angle is an angle with a measure of 90 ° . In a sketch or diagram, 90 ° angle is often noted by placing a small square in the angle.

An acute angle is an angle with a measure less than 90 ° .

An obtuse angle is an angle with a measure greater than 90 ° but less than 180 ° .

All the three types of angles are shown below in the figure:

From the provided figure, it is observed that measure of the angle is less than 90 ° .

Thus, the angle shown in the provided figure is an acute angle.

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## What is a right angle, acute angle, and obtuse angle?

Right angle is half of straight angle.
Acute angle has a smaller measure than right one, while obtuse angle has a bigger measure than right one.
See below for details.

#### Explanation:

An angle is a geometric object formed by two half-lines (rays) with a common endpoint called vertex.

Strictly speaking, there are two angles formed this way: from Ray1 to Ray2 clockwise and counterclockwise.

If Ray1 and Ray2 coincide, the angle between them is called full since it takes a full circle to move Ray1 to Ray2.
The measure of a full angle is, by definition, equal to #360^o# .
Obviously, angles measured at #0^o# , #720^o# etc. look exactly the same as an angle of #360^o# , that is Ray1 and Ray2 coincide.

In this terminology, if Ray1 and Ray2 are located along the same line but directed opposite to each other, they form half of a full angle, this angle is called straight and is measured at #180^o# .

Half of a straight angle would be formed by perpendicular to each other Ray1 and Ray2. This is the right angle, and its measure in degrees is #90^o# .
Acute angles are smaller than right ones and obtuse angles are bigger.

## Practice

Students work as individuals to find their match, after index cards with definitions, examples of angle measures, and pictures of angles within polygons are passed out. They have about a minute to find their group.

Once students establish a group that they belong to, they then work in their group to discuss the meanings of their new math terms. They should be discussing the definitions while matching them to examples within their group. After groups are finished the cards should be collected, shuffled and redistributed so that the activity can be repeated. This will give the students an additional practice of using the definitions and viewing the examples.

I suggest dividing the total number of students by 3 to determine how many cards are needed to evenly divide students.