Exercises from the chapter 3 — Angles

Simplify the following measurements: a) $30^{\circ}70^{\prime}$ · b) $45^{\circ}150^{\prime}$ · c) $65^{\circ}39^{\prime}123^{\prime\prime}$ · d) $110^{\circ}58^{\prime}300^{\prime\prime}$ · e) $30^{\circ}56^{\prime}240^{\prime\prime}$

Determine the sums: a) $30^{\circ}40^{\prime} + 15^{\circ}35^{\prime}$ · b) $10^{\circ}30^{\prime}45^{\prime\prime} + 15^{\circ}29^{\prime}20^{\prime\prime}$

Determine the differences: a) $20^{\circ}50^{\prime}45^{\prime\prime} - 5^{\circ}45^{\prime}30^{\prime\prime}$ · b) $31^{\circ}40^{\prime} - 20^{\circ}45^{\prime}$ · c) $90^{\circ}15^{\prime}20^{\prime\prime} - 45^{\circ}30^{\prime}50^{\prime\prime}$ · d) $90^{\circ} - 50^{\circ}30^{\prime}45^{\prime\prime}$

Determine the products: a) $2 \times (10^{\circ}35^{\prime}45^{\prime\prime})$ · b) $5 \times (6^{\circ}15^{\prime}30^{\prime\prime})$

Determine the divisions: a) $(46^{\circ}48^{\prime}54^{\prime\prime}) \div 2$ · b) $(31^{\circ}32^{\prime}45^{\prime\prime}) \div 3$ · c) $(52^{\circ}63^{\prime}42^{\prime\prime}) \div 5$

Determine the value of $x$ in the following cases:

$\overrightarrow{Oa}$ and $\overrightarrow{Ob}$ are two opposite collinear rays. $\overrightarrow{Oc}$ is any ray. Are angles $a\hat{O}c$ and $c\hat{O}b$ adjacent? Are they supplementary?

Prove: If two angles are vertex-opposed, then they are congruent.

Determine the value of $x$ in the following cases:

Determine the value of $\alpha$ in the following cases:

If $\overrightarrow{OP}$ is angle bisector of $A\hat{O}B$, determine $x$ in the cases:

Are the angles in the following figure complementary? Are they adjacent?

Calculate the value of $x$ in the case below, where $m(r\hat{O}s) = 90^{\circ}$.

The sum of two adjacent angles is $120^{\circ}$. Calculate the measure of each angle, knowing that the measure of one of them is the difference between three times the other and $40^{\circ}$.

Calculate the complement of the following angles: a) $25^{\circ}$ · b) $47^{\circ}$ · c) $37^{\circ}25^{\prime}$

Calculate the supplement of the following angles: a) $72^{\circ}$ · b) $141^{\circ}$ · c) $93^{\circ}15^{\prime}$

Given a measuring angle $x$, indicate: a) its complement; b) its supplement; c) twice its complement; d) half of your supplement; e) triple your supplement; f) the seventh part of the complement; g) the fifth part of the supplement; h) the complement of its third part; i) triple the supplement of its fifth part.

Give the measure of the angle that is twice its complement.

Determine the measure of the angle equal to three times its complement.

Calculate the angle that is four times its complement.

Calculate an angle, knowing that a quarter of your supplement is worth $36^{\circ}$.

What is the angle that exceeds its complement by $76^{\circ}$?

What is the angle that exceeds your supplement by $66^{\circ}$?

Determine an angle, knowing that its supplement exceeds the angle itself by $70^{\circ}$.

What is the angle that added to triple its complement gives $210^{\circ}$?

An angle exceeds its complement by $48^{\circ}$. Determine the supplement of this angle.

The supplement of an angle exceeds this angle by $120^{\circ}$. Determine the angle.

The third’s complement of an angle exceeds the complement of that angle by $30^{\circ}$. Determine the angle.

The supplement of the triple of the half’s complement of an angle is equal to the triple of the complement of that angle. Determine the angle.

The supplement of the complement of an angle exceeds one-third of the complement of the double of that angle in $85^{\circ}$. Determine the angle.

Two angles are supplementary and the ratio between the complement of one and the supplement of the other, in that order, is $\dfrac{1}{8}$. Determine these angles.

Two angles are in the relationship $\dfrac{4}{9}$. Being $130^{\circ}$ its sum, determine the complement of the smallest.

Determine two supplementary angles, knowing that one of them is triple the other.

Two angles are supplementary. One of them is the complement of the fourth part of the other. Calculate these angles.

The ratio between two supplementary angles is equal to $\dfrac{2}{7}$. Determine the complement of the minor.

Determine the complement of an angle, knowing that the ratio between the angle and its complement is equal to $\dfrac{5}{4}$.

The complement of an angle is to its supplement as 2 to 7. Calculate the measure of the angle.

Triple the complement of an angle, increased by $50^{\circ}$, is equal to the supplement of the angle. Determine the measure of the angle.

Determine the measures of two supplementary angles, knowing that twice one of them, added to the seventh part of the other, results in $100^{\circ}$.

The sum of an angle and the third part of its complement results in $46^{\circ}$. Determine the supplement of this angle.

Determine two complementary angles such that twice one, increased by a third of the other, is equal to a right angle.

In the figure, the angle $x$ measures one-sixth of the angle $y$, plus half of the angle $z$. Calculate the angle $y$.

The angles $\alpha$ and $\beta$ are opposite at the vertex. The first is expressed in degrees by $9x - 2$ and the second by $4x + 8$. Determine these angles.

Five rays start from the same point $V$, forming five angles that cover the entire plane and are proportional to the numbers 2, 3, 4, 5 and 6. Calculate the largest angle.

Show that the angle bisectors of two angles opposite the vertex are opposite rays. (Use the “angle bisectors of x and y” button in the o.p.v. angles figure.)

Show that the angle bisectors of two adjacent and supplementary angles form right angles.

Show that the angle bisectors of two adjacent and complementary angles form an angle of $45^{\circ}$.

Two adjacent angles add up to $136^{\circ}$. What is the measure of the angle formed by your angle bisectors?

The angle bisectors of two consecutive angles form an angle of $52^{\circ}$. If one of them measures $40^{\circ}$, what is the measurement of the other?