Question

unit 4: triangles and the value of the pronumerals
question 1 find the value of the pronumeral.
question 2 find the value of the pronumeral.
question 3 find the value of the pronumerals.

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Also, the exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Step2: Solve Question 1a

We know that for a triangle with angles 85°, 65° and \(x\), using the angle - sum property \(85 + 65+x=180\). So \(x = 180-(85 + 65)=30\).

Step3: Solve Question 1b

Since the triangle has two equal sides (indicated by the marks), it is an isosceles triangle. So \(x = 40\) (because \(180-(70 + 70)=40\)).

Step4: Solve Question 1c

For an equilateral triangle, all angles are equal. Let the angle be \(a\). Then \(a+a + a=180\), so \(a = 60\).

Step5: Solve Question 1d

In a right - angled triangle with one angle 30°, using the angle - sum property \(m=180-(90 + 30)=60\).

Step6: Solve Question 1e

Let the third angle of the triangle be \(x\). First, find the non - exterior interior angle adjacent to 135° which is \(180 - 135=45\). Then \(x=180-(25 + 45)=110\).

Step7: Solve Question 1f

In a right - angled triangle with one angle 35°, \(y=180-(90 + 35)=55\).

Step8: Solve Question 2a

The exterior angle 125° is equal to the sum of the two non - adjacent interior angles. So \(x=125 - 45 = 80\).

Step9: Solve Question 2b

The exterior angle \(x\) is equal to \(60+58 = 118\).

Step10: Solve Question 2c

First, find the non - exterior interior angles adjacent to 120° and 100°. The non - exterior angle adjacent to 120° is 60° and adjacent to 100° is 80°. Then \(a=180-(60 + 80)=40\).

Step11: Solve Question 2d

The exterior angle 128° is equal to the sum of the two non - adjacent interior angles. So \(a=128 - 90=38\) (assuming it's a right - angled triangle, if not specified otherwise we can't be sure, but based on common sense of such problems).

Step12: Solve Question 2e

The non - exterior angle adjacent to 120° is 60° and adjacent to 105° is 75°. Then \(x=180-(60 + 75)=45\).

Step13: Solve Question 2f

The exterior angle \(a\) is equal to \(30+70 = 100\).

Step14: Solve Question 3a

For the two right - angled triangles, consider the angle \(a\). The non - right angles of the first right - angled triangle are 60° and 30°. Since the two triangles are related, and using vertical angles and angle - sum property, \(a = 30\) and \(b = 60\).

Step15: Solve Question 3b

The non - exterior angle adjacent to 120° is 60°. Then \(y=180-(95+30)=55\) and \(x = 180-(60 + 55)=65\).

Step16: Solve Question 3c

Using the angle - sum property of a triangle, \(m=180-(75 + 55)=50\). And if the lines are parallel, \(n = 55\) (corresponding angles).

Answer:

Question 1:
a. \(x = 30\)
b. \(x = 40\)
c. \(a = 60\)
d. \(m = 60\)
e. \(x = 110\)
f. \(y = 55\)
Question 2:
a. \(x = 80\)
b. \(x = 118\)
c. \(a = 40\)
d. \(a = 38\)
e. \(x = 45\)
f. \(a = 100\)
Question 3:
a. \(a = 30\), \(b = 60\)
b. \(x = 65\), \(y = 55\)
c. \(m = 50\), \(n = 55\)