Problem 1 :
In the figure given below, let the lines l₁ and l₂ be parallel and m is transversal. If ∠F = 65°, find the measure of each of the remaining angles.
Problem 2 :
Can 30°, 60° and 90° be the angles of a triangle ?
Problem 3 :
In a triangle, if the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.
Problem 4 :
Find m∠W and m∠X in the triangle given below.
1. Answer :
From the given figure,
∠F and ∠H are vertically opposite angles and they are equal.
Then,
∠H = ∠F ----> ∠H = 65°
∠H and ∠D are corresponding angles and they are equal.
Then,
∠D = ∠H ----> ∠D = 65°
∠D and ∠B are vertically opposite angles and they are equal.
Then,
∠B = ∠D ----> ∠B = 65°
∠F and ∠E are together form a straight angle.
Then, we have
∠F + ∠E = 180°
Substitute ∠F = 65°.
∠F + ∠E = 180°
65° + ∠E = 180°
∠E = 115°
∠E and ∠G are vertically opposite angles and they are equal.
Then,
∠G = ∠E ----> ∠G = 115°
∠G and ∠C are corresponding angles and they are equal.
Then,
∠C = ∠G ----> ∠C = 115°
∠C and ∠A are vertically opposite angles and they are equal.
Then,
∠A = ∠C ----> ∠A = 115°
Therefore,
∠A = ∠C = ∠E = ∠G = 115°
∠B = ∠D = ∠F = ∠H = 65°
2. Answer :
Let us add all the three given angles and check whether the sum is equal to 180°.
30° + 60° + 90° = 180°
Since the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.
3. Answer :
Let x be the first angle.
The second angle = x + 5.
The third angle = x + 5 + 5 = x + 10.
We know that,
the sum of the three angles of a triangle = 180°
x + (x + 5) + (x + 10) = 180
3x + 15 = 180
3x = 165
x = 55
The first angle = 55°.
The second angle = 55 + 5 = 60°.
The third angle = 60 + 5 = 65°.
So, the three angles of a triangle are 55°, 60° and 65°.
4. Answer :
Step 1 :
Write the Exterior Angle Theorem as it applies to this triangle.
m∠W + m∠X = m∠WYZ
Step 2 :
Substitute the given angle measures.
(4y - 4)° + 3y° = 52°
Step 3 :
Solve the equation for y.
(4y - 4)° + 3y° = 52°
4y - 4 + 3y = 52
Combine the like terms.
7y - 4 = 52
Add 4 to both sides.
7y - 4 + 4 = 52 + 4
Simplify.
7y = 56
Divide both sides by 7.
y = 8
Step 4 :
Use the value of y to find m∠W and m∠X.
m∠W = 4y - 4
= 4(8) - 4
= 32 - 4
= 28
m∠X = 3y
= 3(8)
= 24
So, m∠W = 28° and m∠X = 24°.
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