Question

2 lines intersect. Where the 2 lines intersect, 4 angles are created. Labeled clockwise, from uppercase right: angle 1 (3 x minus 1) degrees, angle 2 is blank, angle 3 (2 x + 9) degrees, and angle 4 is blank.
What are the numerical measures of each angle in the diagram?
∠1 and ∠3 measure degrees.
∠2 and ∠4 measure degrees.

Answer 1

Firstly, vertical angles can be defined as a pair of opposite angles formed by intersecting lines.

∴ They are always congruent to each other

As per the diagram below,

∠1 = ∠3 [∵ vertical angles are always equal] ...(Eqn. 1)

Similarly, ∠2 = ∠4

Now;

∠1 + ∠2 = 180° (linear pair) ...(Eqn. 2)

Likewise, ∠3 + ∠4 = 180°

According to the question;

∠1 = $(3x-1)$°

∠3 = $(2x + 9)$°

On substituting these values in Eqn. 1, we get,

$(3x - 1) = (2x +9)$

⇒ $3x - 2x = 9 + 1$

⇒ $x = 10$

∴ Measure of ∠1 = $(3x - 1)$° $= (3$ × $10 - 1)$° $= 29$°

Similarly, measure of ∠3 = $29$° (as per Eqn. 1)

Now, for finding the values of ∠2 and ∠4 respectively, we have to substitute the value of ∠1 in Eqn. 2;

∠1 + ∠2 = 180°

∴ 29° + ∠2 = 180°

⇒ ∠2 = (180 - 29)°

⇒ ∠2 = 151°

So, the measure of ∠2 = 151°

Then, the measure of ∠ 4 = 151° (since ∠2 = ∠4)

Ans) ∠1 and ∠3 measure 29° each. ∠2 and ∠4 measure 151° each.