Points, Lines, and Angles: Additional Practice (Set 1)

A linear pair is supplementary, so the two angles sum to $180^\circ$.

$(4x - 20) + x = 180 \implies 5x = 200 \implies x = 40$

Check: the angles are $4(40) - 20 = 140^\circ$ and $40^\circ$, which add to $180^\circ$.

Supplementary angles sum to $180^\circ$. Let the smaller be $x$; the larger is $3x - 40$.

$x + (3x - 40) = 180 \implies 4x = 220 \implies x = 55$

The smaller angle is $55^\circ$ (the larger is $3(55) - 40 = 125^\circ$).

Vertical angles are equal, so set the two expressions equal.

$2x + 30 = 3x - 10 \implies x = 40$

Each angle is $2(40) + 30 = 110^\circ$ (check: $3(40) - 10 = 110^\circ$).

Complementary angles sum to $90^\circ$. Write the angles as $4k$ and $5k$.

$4k + 5k = 90 \implies 9k = 90 \implies k = 10$

The larger angle is $5(10) = 50^\circ$ (the smaller is $40^\circ$).

The opening and its reflex angle together make one full turn of $360^\circ$.

$360^\circ - 145^\circ = 215^\circ$

Since $215^\circ$ is between $180^\circ$ and $360^\circ$, it is indeed a reflex angle.

Angles around a point sum to $360^\circ$.

$x + (x + 40) + 2x = 360 \implies 4x + 40 = 360 \implies x = 80$

The three angles are $80^\circ$, $120^\circ$, and $2(80) = 160^\circ$, so the largest is $160^\circ$.

Equal angles that are supplementary each equal $x$, and they sum to $180^\circ$.

$x + x = 180 \implies x = 90$

Each angle is $90^\circ$, a right angle.

Both pairs are complementary, so each of $A$ and $C$ equals $90^\circ$ minus angle $B$.

$A = 90^\circ - B \quad\text{and}\quad C = 90^\circ - B \implies A = C$

Two angles complementary to the same angle are equal.

When parallel lines are cut by a transversal, alternate interior angles are equal.

$5x - 20 = 3x + 10 \implies 2x = 30 \implies x = 15$

Each angle is $5(15) - 20 = 55^\circ$ (check: $3(15) + 10 = 55^\circ$).

The angle across from the $90^\circ$ angle is its vertical angle, so it is also $90^\circ$. Each of the other two is a linear pair with a $90^\circ$ angle.

$180^\circ - 90^\circ = 90^\circ$

So all four angles are $90^\circ$, which is why perpendicular lines are said to meet at right angles.

Angles around a point sum to $360^\circ$. Write them as $2k$, $3k$, and $7k$.

$2k + 3k + 7k = 360 \implies 12k = 360 \implies k = 30$

The largest angle is $7(30) = 210^\circ$ (a reflex angle).

Co-interior (same-side interior) angles between parallel lines are supplementary, so they sum to $180^\circ$.

$(2x + 10) + (3x - 15) = 180 \implies 5x - 5 = 180 \implies x = 37$

The angles are $2(37) + 10 = 84^\circ$ and $3(37) - 15 = 96^\circ$, so the larger is $96^\circ$.