Envision Math Grade 8 Volume 1 Chapter 6 Congruence And Similarity
Page 353 Exercise 2 Answer
Given:-Two parallel lines and transversals lines
Find out:-Write the properties of parallel lines and transversals lines.
If we have two parallel lines and transversals lines then we use the properties:-
Alternate Interior angles are equal.
Alternate exterior angles are equal.
The Sum of angles made on the same sides of the transversals is 180∘.
Corresponding angles are equal.
We use the four properties of parallel lines and transversals lines to solve the problems.
Page 353 Focus On Math Practices Answer
Given: The measurements of the two angles are 65∘.
To find m∠1 using assumption and explain the assumption reason.
It is given that the measurements of the two angles are 65∘. Therefore, x is:
The assumption made to find m∠1 is to subtract the sum of the two given angle measures 65 degrees and 65 degrees from 180 degrees. It is reasonable because the given tiles are triangular and the sum of the angles in the triangle is 180 degrees.
Page 354 Try It Answer
Given:- ∠2 = 68∘ and ∠3 = 40∘
Find out:- ∠1 = ?
Use the angle sum property of a triangle.
Thus, the measure of unknown angle is 72∘
The measure of unknown angle is 72∘
Page 354 Convince Me Answer
Given:- angle measures of a triangle 23∘,71∘ and 96∘
Explain:- these angle measures can be possible or not?
We use here angle sum property of a triangle, i.e. the sum of all interior angles is 180∘.
So,
23∘ + 71∘ + 96∘ = 190∘
This is not possible for a triangle.
A triangle could not have interior angle measures of 23∘,71∘ and 96∘
Page 354 Convince Me Answer
Given:- angle measures of a triangle 23∘,71∘ and 96∘
Explain:- these angle measures can be possible or not?
We use here angle sum property of a triangle, i.e. the sum of all interior angles is 180∘.
So,
23∘ + 71∘ + 96∘ = 190∘
This is not possible for a triangle.
A triangle could not have interior angle measures of 23∘,71∘ and 96∘
Page 356 Exercise 2 Answer
Given:- ∠1 = 90∘ and ∠2 = ∠3
Find out:- all the possible values of exterior angles.
Find the values of angles 2 and 3 using angle sum property, then find the measure for the exterior angles by using exterior angle property for a triangle.
The triangle can be represented as follows:
Given that ∠1 = 90∘ and ∠2 = ∠3
Therefore, using angle sum property of the triangle,
∠1 + ∠2 + ∠3 = 180∘
Now the exterior angles 4, 5 and 6 can be found using the exterior angle property:
∴ ∠6 = 45° + 45°, ∠5 = 90° + 45° and ∠4 = 90° + 45°
∠6 = 90°, ∠5 = 135° and ∠4 = 135°
These are all the possible exterior angles for this triangle.
All possible values of the exterior angle are ∠6 = 90∘, ∠5 = 135∘ and ∠4 = 135∘
Page 356 Exercise 3 Answer
Given:- two angles are 32∘ and 87∘ and one exterior angle is 93∘
Find out:- all the interior angles and exterior angles of this triangle.
use the angle sum property of a triangle and exterior angle property.
By using the angle sum property of a triangle,
87∘ + 32∘ + ∠3 = 180∘
∠3 = 180∘ − 119∘
∠3 = 61∘
and ∠4 = 87∘ + 32∘ [exterior angle property of a triangle]
∠4 = 119∘
and ∠6 = 87∘ + 61∘ [exterior angle property of a triangle]
∠6 = 148∘
Hence, the diagram of the triangle can be drawn as:
The remaining interior angle is 61∘ and exterior angles are 119∘ and 148∘
The final diagram of the triangle is:
Page 356 Exercise 4 Answer
Given:- a ∥ b and some angles in the given diagram.
find out:- ∠1 and ∠2.
we use properties of parallel lines and transversals and linear pairs.
as
a ∥ b
∠2 = 37.3∘ [alternate interior angles]
and ∠1 + 79.4∘ + 37.3∘ = 180∘ [Co−interior angles]
∠1 = 180∘ − 116.7∘
∠1 = 63.3∘
The angles are ∠1 = 63.3∘ and ∠2 = 37.3∘
Page 356 Exercise 5 Answer
Given ∥ b and some angles in the given diagram.
find out:∠3 and ∠4 =?
we use properties of parallel lines and transversals and linear pairs.
The angles are ∠3 = 63.3∘ and ∠4 = 142.7∘
Page 356 Exercise 6 Answer
Given ΔABC, m ∠A = x∘, m B = (2x)∘, m ∠C = (6x+18)∘
Find the measure of each angle.
Use the triangle angle sum theorem to find each angle.
The angle measures are: m ∠A = 18∘, m ∠B = 36∘, m ∠C = 126∘
Page 357 Exercise 7 Answer
Given the figure of the triangle.
Use the exterior angle theorem to find the required answer.
Here in the figure, angle ∠1 is the exterior angle of the triangle.
m∠1 is equal to the sum of two remote interior angles.
According to the exterior angle theorem,
∠1 = 59∘ + 56∘
∠1 = 115∘
∠1 is the exterior angle of the triangle.
m∠1 is equal to the sum of two remote interior angles.
m∠1 = 115∘
Page 357 Exercise 9 Answer
Given the figure.
Use the triangle angle sum theorem to find the required angle.
The required angle is m ∠C = 83.5∘.
Page 357 Exercise 10 Answer
Given the figure.
Use the exterior angle theorem to find the required answer.
Notice that, ∠4 is an exterior angle of the given triangle. Moreover, angle 1 and ∠2 are its remote interior angles. Therefore, we can use the fact that the measure of an exterior angle of a triangle is, equal to the sum of the measures of its remote interior angles and calculate the value of x.
Our friend got the wrong measure of ∠4 because
he probably think that m∠1, m∠2 and m∠4 have the sum of 180∘
Therefore he got m∠4 = 51∘, which is the measure of ∠3, the third interior angle of the given triangle.
do not have the sum of 180∘
because, as we can see in the picture, not all three angles are interior.
The measure of the angle is ∠4 = 129∘.
Page 358 Exercise 12 Answer
Given the figure of the triangle.
Use the exterior angle theorem to find the required answer.
According to the exterior angle theorem,
m∠4 = m∠2 + m∠1
The value of m∠3 is calculated as:
m∠3 = 161 − 25 × 2
= 161 − 50
= 111∘
The expression for m∠3 = 161 − 25x and the value of m∠3 = 111∘
Page 358 Exercise 13 Answer
Given a figure of a triangle.
Use the triangle angle sum theorem to find the required angle.
The measure of the acute angle is x = 52.8∘.
Page 358 Exercise 14 Answer
Given a figure of a triangle.
Use the exterior angle theorem to find the required angles.
Here in the figure, ∠A,∠B,∠C are interior angles. Angle ∠C is adjacent angle whereas ∠A,∠B are non-adjacent angles.
Therefore the remote interior angles for an exterior angle ∠F is ∠A and ∠B.
The two remote interior angles for an exterior angle ∠F is ∠A and ∠B.
Page 358 Exercise 15 Answer
Given a figure of a triangle.
Use the exterior angle theorem to find the required angle.
The value of m∠3 = 7x + 10 = 7 × 20 + 10 = 150
The value of m∠3 = 150∘.