QuanticaPrealgebraOpen in the courseGeometry begins with the plainest things you can imagine. A single point. Two points joined make a segment, stretch it one way and you get a ray, stretch it both ways and you get a line. Now let two rays start from the same point. They open up, and the space between them is an angle. In this lesson you will learn to name angles and to measure how far they open, counted in degrees.
Problem
In the diagram above, \(\angle ABC\) is a right angle (the small square at \(B\) marks it), and ray \(BX\) splits it into two parts. The diagram gives \(\angle XBC = 28^\circ\). Find \(\angle ABX\). Give the number of degrees.
Show a hint
- The little square tells you \(\angle ABC\) is a right angle, so the whole thing measures \(90^\circ\).
- Ray \(BX\) cuts that right angle into \(\angle ABX\) and \(\angle XBC\), and those two parts add up to the full \(90^\circ\). You know one part already.
Show the full solution
The small square means \(\angle ABC\) is a right angle, so the two pieces ray \(BX\) makes must fill \(90^\circ\). One piece is given, so the other is what's left over. $$\angle ABX = 90^\circ - 28^\circ = \boxed{62}$$
Problem
In the diagram above, \(A\), \(O\), \(B\) lie on a straight line and ray \(OC\) stands on it. The diagram gives \(\angle AOC = 124^\circ\). Find \(\angle BOC\). Give the number of degrees.
Show a hint
- \(\angle AOC\) and \(\angle BOC\) sit side by side along the straight line \(AB\), so together they form a straight angle. How many degrees is a straight angle?
- A straight angle is \(180^\circ\). Subtract the part you already know from \(180\) to get the part you want.
Show the full solution
\(\angle AOC\) and \(\angle BOC\) sit side by side along the straight line \(AB\), so together they make a straight angle of \(180^\circ\). $$\angle BOC = 180 - 124 = \boxed{56}$$
Problem
In the diagram above, ray \(OB\) lies inside \(\angle AOC\), and \(\angle AOB = 88^\circ\). The whole angle \(\angle AOC = 147^\circ\). What is \(\angle BOC\)? Give the number of degrees.
Show a hint
- Ray \(OB\) cuts the big angle into two pieces, and those two pieces add up to the whole \(\angle AOC\).
- You know the whole angle and one piece, so \(\angle BOC\) is what is left over. Subtract.
Show the full solution
The two parts \(\angle AOB\) and \(\angle BOC\) add up to the whole angle \(\angle AOC\). So \(\angle BOC\) is the whole minus the part you already know. $$\angle BOC = 147^\circ - 88^\circ = \boxed{59}$$
Problem
In the diagram above, \(\angle AOC\) is a right angle and ray \(OD\) splits it. Given that \(\angle AOD = 37^\circ\), find \(\angle DOC\). Give the number of degrees.
Show a hint
- A right angle measures \(90^\circ\), and ray \(OD\) breaks it into the two pieces \(\angle AOD\) and \(\angle DOC\).
- Those two pieces together make the whole right angle, so \(\angle AOD + \angle DOC = 90^\circ\). Subtract to find the missing piece.
Show the full solution
Ray \(OD\) splits the right angle \(\angle AOC\) into two parts that add to \(90^\circ\). The two angles are complementary, so $$\angle DOC = 90 - 37 = \boxed{53}.$$
Problem
In the diagram above, two rays make an angle of \(110^\circ\), and the question mark marks the reflex angle, the big way around the outside. What is the reflex angle? Give the number of degrees.
Show a hint
- A full turn all the way around a point is \(360^\circ\). The reflex angle and the \(110^\circ\) angle together sweep out that whole turn.
- So the reflex angle is what is left after you take the \(110^\circ\) away from \(360^\circ\).
Show the full solution
The \(110^\circ\) angle and the reflex angle together sweep the full turn around the vertex, which is \(360^\circ\). $$360^\circ - 110^\circ = \boxed{250}$$ An angle and the reflex angle beside it always pair up to \(360^\circ\), so either one hands you the other.
Problem
In the diagram above, ray \(OQ\) bisects \(\angle POR\), so the equal tick marks show the two halves are equal. Given that one half \(\angle POQ = 41^\circ\), what is the whole \(\angle POR\)? Give the number of degrees.
Show a hint
- A bisector cuts an angle into two equal halves, so \(\angle POR\) is made of two copies of \(\angle POQ\).
- The whole is twice one half. Double \(41^\circ\).
Show the full solution
A bisector splits an angle into two equal halves, so the whole is twice one half. $$\angle POR = 2 \times 41^\circ = \boxed{82}^\circ$$
Problem
In the diagram above, two straight lines cross at a point, forming four angles. One of them measures \(74^\circ\), and the angle marked with the question mark sits directly across from it. What is the marked angle? Give the number of degrees.
Show a hint
- The two marked angles are directly across the crossing point from each other. What is that pair of angles called?
- Vertical angles, the pair sitting opposite each other where two lines cross, are always equal. So the marked angle matches the \(74^\circ\) one.
Show the full solution
The marked angle sits directly across the crossing point from the \(74^\circ\) angle, so the two are vertical angles and have the same measure. The marked angle is \(\boxed{74}\) degrees. Vertical angles match because each one shares a straight line with the same neighbor, so both come out as \(180^\circ\) minus that neighbor.
Problem
In the diagram above, three rays leave a single point, cutting the whole way around into three angles. Two of them measure \(137^\circ\) and \(95^\circ\). Find the third angle, the one marked with the question mark. Give the number of degrees.
Show a hint
- Going all the way around a point is one full turn, and a full turn is \(360^\circ\). The three angles have to add up to that.
- So the missing angle is what's left after you take the two known ones away from \(360^\circ\).
Show the full solution
The three angles wrap all the way around the point, so they add to a full turn of \(360^\circ\). Subtract the two you know. $$360 - 137 - 95 = \boxed{128}$$
Problem
In the diagram above, a ray splits the right angle \(\angle ABC\) into two parts whose measures are in the ratio \(1 : 5\). Find the smaller part, the one marked with the question mark. Give the number of degrees.
Show a hint
- A right angle is \(90^\circ\), and the two parts together make up that whole \(90^\circ\).
- A ratio of \(1 : 5\) means the angle is cut into \(1 + 5 = 6\) equal shares. The smaller part is just one of those shares.
Show the full solution
The ratio \(1 : 5\) splits the right angle into \(1 + 5 = 6\) equal shares. Since a right angle is \(90^\circ\), each share is $$\frac{90^\circ}{6} = 15^\circ.$$ The smaller part is one share, so it measures \(\boxed{15}\) degrees.
Problem
The clock in the diagram above shows 5:00. Find the measure of the (non-reflex) angle between the hour hand and the minute hand. Give the number of degrees.
Show a hint
- The 12 hour marks split the full \(360^\circ\) evenly. How many degrees is one gap between neighboring marks?
- Count how many of those gaps sit between the two hands at 5:00, then add up that many equal pieces.
Show the full solution
The 12 marks divide the full circle evenly, so each gap is \(\frac{360^\circ}{12} = 30^\circ\). At 5:00 the minute hand points at 12 and the hour hand points at 5, which is 5 gaps away. So the angle is \(5 \times 30^\circ = \boxed{150}\) degrees.
Problem
In the diagram above, a ray stands on a straight line, and the two angles it forms are labeled \(x^\circ\) and \((x+10)^\circ\). Find \(x\). Give the number of degrees (the value of \(x\)).
Show a hint
- The ray sits on a straight line, so the two angles together open up all the way across it. What does a straight angle measure?
- The two labeled angles are supplementary, so they add to \(180^\circ\). Set up \(x + (x+10) = 180\) and solve for \(x\).
Show the full solution
A ray standing on a straight line makes two supplementary angles, so they add to \(180^\circ\). Adding the labels gives $$x + (x+10) = 180,$$ which simplifies to \(2x + 10 = 180\). Then \(2x = 170\), so \(x = \boxed{85}\).
Problem
In the diagram above, two rays split the straight angle into three angles whose measures are in the ratio \(2 : 3 : 4\). Find the largest of the three, the one marked with the question mark. Give the number of degrees.
Show a hint
- A straight angle measures \(180^\circ\), and the three parts together fill it. Think of the ratio \(2 : 3 : 4\) as splitting that \(180^\circ\) into equal shares.
- Add the ratio numbers: \(2 + 3 + 4 = 9\). So \(180^\circ\) is cut into \(9\) equal shares. The largest angle takes \(4\) of them.
Show the full solution
The three angles sit along a straight line, so they add to \(180^\circ\). The ratio \(2 : 3 : 4\) cuts that into \(2 + 3 + 4 = 9\) equal shares, so each share is \(\frac{180^\circ}{9} = 20^\circ\). The largest angle is \(4\) shares.
$$\frac{4}{9} \times 180^\circ = 4 \times 20^\circ = \boxed{80}$$
Problem
In the diagram above, an angle \(x^\circ\) and its supplement \((180-x)^\circ\) sit side by side on a straight line. The angle is \(15^\circ\) more than twice its supplement. Find the angle. Give the number of degrees.
Show a hint
- Two angles that sit together on a straight line are supplementary, so they add to \(180^\circ\). That is why the supplement is written \((180-x)^\circ\).
- Turn the words into an equation. "\(15^\circ\) more than twice its supplement" means \(x = 2(180-x) + 15\). Expand the right side and collect the \(x\) terms.
Show the full solution
The two angles sit together on a straight line, so the supplement of \(x^\circ\) is \((180-x)^\circ\). The angle is \(15^\circ\) more than twice that, so $$x = 2(180 - x) + 15.$$ Expanding gives \(x = 360 - 2x + 15\), so \(3x = 375\) and \(x = \boxed{125}\).
Practice these ideas
Practice
In the diagram above, \(\angle ABC\) is a right angle and ray \(BX\) splits it into two smaller angles. Given that \(\angle XBC = 34^\circ\), what is \(\angle ABX\)? Give the number of degrees.
Show the solution
The two pieces sit side by side and fill the right angle, so they add to \(90^\circ\). That means \(\angle ABX\) is whatever is left after taking away \(\angle XBC\). $$\angle ABX = 90^\circ - 34^\circ = \boxed{56}$$
Practice
In the diagram above, ray \(OB\) lies inside \(\angle AOC\), splitting it into two smaller angles. The diagram shows \(\angle AOB = 92^\circ\), and the whole \(\angle AOC = 155^\circ\). What is \(\angle BOC\)? Give the number of degrees.
Show the solution
\(\angle AOB\) and \(\angle BOC\) are the two pieces of \(\angle AOC\), so the missing piece is the whole minus the part you know. $$\angle BOC = 155^\circ - 92^\circ = \boxed{63}$$
Practice
In the diagram above, \(AOB\) is a straight line and ray \(OC\) rises from point \(O\). Given that \(\angle AOC = 113^\circ\), what is \(\angle BOC\)? Give the number of degrees.
Show the solution
Angles on a straight line add up to \(180^\circ\). Since \(\angle AOC\) and \(\angle BOC\) together form the straight line \(AOB\),
$$\angle BOC = 180^\circ - 113^\circ = \boxed{67}^\circ.$$
Practice
In the diagram above, \(\angle AOC\) is a right angle and ray \(OD\) falls inside it. Given that \(\angle AOD = 29^\circ\), what is \(\angle DOC\)? Give the number of degrees.
Show the solution
The two smaller angles sit inside the right angle and add up to it, so \(\angle AOD + \angle DOC = 90^\circ\). Subtract the part you know. $$\angle DOC = 90^\circ - 29^\circ = \boxed{61}$$
Practice
In the diagram above, ray \(OQ\) bisects \(\angle POR\), so the two halves are equal. Given that \(\angle POQ = 38^\circ\), what is the whole \(\angle POR\)? Give the number of degrees.
Show the solution
A bisector cuts an angle into two equal halves, so the whole is twice one half. $$\angle POR = 2 \times 38^\circ = \boxed{76}$$
Practice
In the diagram above, two lines cross and one of the four angles measures \(116^\circ\). The marked angle sits right beside it along the same straight line. What is the measure of the marked angle? Give the number of degrees.
Show the solution
The marked angle and the \(116^\circ\) angle sit side by side along a straight line, so they add to \(180^\circ\). $$180^\circ - 116^\circ = \boxed{64}$$ Where two lines cross, every angle is either equal to a given one or supplementary to it.
Practice
In the diagram above, three rays from a single point split the space all the way around into three angles. Two of them measure \(88^\circ\) and \(145^\circ\). What is the third angle? Give the number of degrees.
Show the solution
The three angles fill a complete turn around the point, so they add to \(360^\circ\). Subtract the two you know. $$360 - 88 - 145 = \boxed{127}$$
Practice
In the diagram above, two rays open up to make an angle of \(145^\circ\). The question mark marks the reflex angle, the one that sweeps all the way around the outside. What is the reflex angle? Give the number of degrees.
Show the solution
The two angles at the vertex fill one complete turn, so together they add to \(360^\circ\). Subtract the part you know from the whole turn. $$360^\circ - 145^\circ = \boxed{215}^\circ$$
Practice
In the diagram above, ray \(OP\) stands on a straight line, splitting the straight angle into \(x^\circ\) and \(2x^\circ\). Find \(x\). Give the number of degrees.
Show the solution
The two angles rest on a straight line, so together they form a straight angle of \(180^\circ\). Adding the pieces gives $$x + 2x = 3x = 180,$$ so \(x = \boxed{60}\).
Practice
In the diagram above, ray \(BX\) splits the right angle \(\angle ABC\) into two parts whose measures are in the ratio \(2 : 7\). What is the measure of the smaller part? Give the number of degrees.
Show the solution
The ratio \(2 : 7\) cuts the right angle into \(2 + 7 = 9\) equal shares, so each share is \(90^\circ \div 9 = 10^\circ\). The smaller part is \(2\) shares. $$\frac{2}{9} \times 90^\circ = \boxed{20}$$
Practice
In the diagram above, the angle \(y^\circ\) and its complement \((90-y)^\circ\) together fill a right angle. The angle exceeds its complement by \(40^\circ\). What is \(y\)? Give the number of degrees.
Show the solution
The angle is \(40^\circ\) more than its complement, so $$y = (90 - y) + 40.$$ The right side is \(130 - y\), so \(2y = 130\) and \(y = \boxed{65}\). The complement is then \(25^\circ\), and \(65 - 25 = 40\) checks out.
Practice
The clock in the diagram above reads 3:00. What is the non-reflex angle between the hour hand and the minute hand? Give the number of degrees.
Show the solution
The clock face is a full turn of \(360^\circ\) divided into 12 equal hour steps, so each step is \(360^\circ \div 12 = 30^\circ\). At 3:00 the hour hand points at 3 and the minute hand points at 12, which are 3 steps apart. So the angle is \(3 \times 30^\circ = \boxed{90}^\circ\).
Practice
In the diagram above, three rays from a single point split the full turn around that point into three angles whose measures are in the ratio \(3 : 4 : 5\). What is the measure of the largest of these three angles? Give the number of degrees.
Show the solution
The three angles go all the way around the point, so they add to a full turn of \(360^\circ\). The ratio \(3 : 4 : 5\) splits that turn into \(3 + 4 + 5 = 12\) equal shares, so one share is \(\frac{360^\circ}{12} = 30^\circ\). The largest angle is \(5\) shares. $$5 \times 30^\circ = \frac{5}{12} \times 360^\circ = \boxed{150}$$