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Points, Lines, and Angles: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 The marked opening

Write both three-letter names for the opening marked by the arc in the figure.

Three rays from $Q$ , with one arc marking an opening.

Text description of this figure

Three rays share the endpoint Q. One goes horizontally to the right through point P, one rises to the upper right through point R, and one goes straight up through point S; each ray has an arrowhead beyond its labeled point. A single arc near Q starts on ray QP, passes across ray QR, and ends on ray QS, marking the whole opening between the two outer rays. No angle measures are shown.
1. Hint 1
  An angle name identifies its two sides and their shared endpoint.
2. Hint 2
  Put the shared endpoint in the middle, and use a point on each side of the marked opening.
Answer

$\angle PQS$ and $\angle SQP$ .

Full solution

The marked opening has sides through $P$ and $S$ .
Both sides start at $Q$ , so $Q$ must be the middle letter.

Reading the two sides in either order gives $\angle PQS$ or $\angle SQP$ .
The point $R$ is on the ray inside the opening, so it does not name either outer side.

Answer

$\angle PQS$ and $\angle SQP$ .

Key idea

The middle letter of a three-letter angle name is its vertex, and the other letters lie on its two sides.
Problem 1: Mark complete
Problem 2 A pointer turn

A pointer turns through $\frac{5}{8}$ of a full turn. Classify the angle it turns through as acute, right, obtuse, straight, or reflex.
1. Hint 1
  Compare the turn with a half turn and a full turn.
2. Hint 2
  A full turn measures $360^\circ$ ; multiply this by the given fraction.
Answer

Reflex.

Full solution

Find the amount of turn in degrees.
$\frac{5}{8}\times360^\circ=225^\circ$

This is greater than $180^\circ$ and less than $360^\circ$ , so the angle is reflex.
The fraction is greater than one-half and less than one, which agrees with that classification.

Answer

Reflex.

Key idea

A turn greater than a half turn and less than a full turn forms a reflex angle.
Problem 2: Mark complete
Problem 3 Points on the drawing

List every group of three labeled points that is collinear in the figure.

Two lines crossing at $B$ , with five labeled points.

Text description of this figure

Two straight lines with arrowheads at both ends cross at point B. The horizontal line carries points A, B and C from left to right, with A and C equally far from B. The other line slants up from lower left to upper right; point D lies on it below and to the left of B, and point E lies on it above and to the right of B, equally far from B. No lengths or angles are marked.
1. Hint 1
  Collinear points lie on one straight line.
2. Hint 2
  Follow each complete line through the crossing and collect its three point labels.
Answer

$A,B,C$ and $D,B,E$ (in any order within each group).

Full solution

The horizontal line contains $A$ , $B$ , and $C$ .
The slanted line contains $D$ , $B$ , and $E$ .
Each of these groups lies on one straight line.

Any other group has a point from the horizontal line other than $B$ and a point from the slanted line other than $B$ .
Two different lines share only the point $B$ , so no single line holds all three.
Therefore the complete list is $A,B,C$ and $D,B,E$ , in any order within each group.

Answer

$A,B,C$ and $D,B,E$ (in any order within each group).

Key idea

To identify collinear points, follow one straight line through all the points in a proposed group.
Problem 3: Mark complete
Problem 4 A line, a ray and a segment

For the figure, write a name for the complete line, the drawn ray, and the segment ending at $H$ , and give each object's endpoints.

A line, a ray and a segment, with five named points.

Text description of this figure

A horizontal line with arrowheads at both ends passes through points D, E and F, in that order from left to right. From point E, a vertical ray goes straight up through point G, with an arrowhead above G. A horizontal segment with solid dots at its ends and no arrowheads joins G to point H on its right. H sits directly above F, but H and F are not joined. No lengths or angles are marked.
1. Hint 1
  Arrowheads show which directions continue beyond the drawing.
2. Hint 2
  Use the endpoint first when naming a ray; a line has no endpoints, and a segment has two.
Answer

Line $\overleftrightarrow{DF}$ (or any two of $D$ , $E$ , $F$ , in either order), no endpoints; ray $\overrightarrow{EG}$ , endpoint $E$ ; segment $\overline{GH}$ or $\overline{HG}$ , endpoints $G,H$ .

Full solution

The line extends in both directions through $D$ , $E$ , and $F$ , so it has no endpoints.
Any two of these three points name it, such as $\overleftrightarrow{DF}$ .

The upward ray starts at $E$ and passes through $G$ , giving $\overrightarrow{EG}$ with endpoint $E$ .
The horizontal piece from $G$ to $H$ has no arrows, so it is $\overline{GH}$ with endpoints $G$ and $H$ .

Answer

Line $\overleftrightarrow{DF}$ (or any two of $D$ , $E$ , $F$ , in either order), no endpoints; ray $\overrightarrow{EG}$ , endpoint $E$ ; segment $\overline{GH}$ or $\overline{HG}$ , endpoints $G,H$ .

Key idea

The arrowheads and stopping points in a drawing determine whether a straight object is a line, ray, or segment.
Problem 4: Mark complete
Problem 5 An opening in two parts

The figure shows an opening split into two parts. Find the complement and supplement of the whole opening $\angle KLM$ , and classify each of those two results as acute, right, obtuse, straight, or reflex.

The opening at $L$ , split by ray $LN$ into two marked parts.

Text description of this figure

Three rays start at point L, each with an arrowhead beyond its labeled point. Ray LK goes horizontally to the right through point K. Ray LN turns upward from LK through point N, and the arc between LK and LN is labeled 21 degrees. Ray LM turns upward further through point M, and the arc between LN and LM is labeled 27 degrees. The angles are drawn to scale, and the whole opening from LK to LM has no measure written on it.
1. Hint 1
  The whole opening includes both marked parts.
2. Hint 2
  Add the two parts before subtracting from $90^\circ$ for the complement or $180^\circ$ for the supplement.
3. Hint 3
  Compare each resulting measure with $90^\circ$ and $180^\circ$ .
Answer

Complement: $42^\circ$ , acute. Supplement: $132^\circ$ , obtuse.

Full solution

The two non-overlapping parts fill the whole opening.
$\angle KLM=21^\circ+27^\circ$ $\angle KLM=48^\circ$

Subtract the whole opening from each required total.
$90^\circ-48^\circ=42^\circ$ $180^\circ-48^\circ=132^\circ$

The complement is acute because it lies between $0^\circ$ and $90^\circ$ .
The supplement is obtuse because it lies between $90^\circ$ and $180^\circ$ .
Checking, $48^\circ+42^\circ=90^\circ$ and
$48^\circ+132^\circ=180^\circ.$

Answer

Complement: $42^\circ$ , acute. Supplement: $132^\circ$ , obtuse.

Key idea

Find the entire angle before calculating its complement or supplement.
Problem 5: Mark complete
Problem 6 Across the two rails

In the figure, lines $a$ and $b$ are parallel. Find the angles marked $u$ , $v$ , and $w$ . For each one, name its relationship to the whole opening split into $28^\circ$ and $43^\circ$ at the upper crossing.

Parallel lines $a$ and $b$ cut by one crossing line.

Text description of this figure

Two horizontal lines, a on top and b below, have arrowheads at both ends and matching parallel marks. One straight line rises from lower left to upper right and crosses both. At the upper crossing, a short extra ray drawn inside the lower-left opening splits it in two: the part next to the leftward direction of line a is marked 28 degrees, and the part next to the downward-left direction of the crossing line is marked 43 degrees. At the lower crossing, three angles have their own arcs and letters: u is the lower-left angle, v is the upper-right angle, and w is the upper-left angle. The lower-right angle at the lower crossing is not marked.
1. Hint 1
  First find the full angle at the upper crossing, then compare positions at the two crossings.
2. Hint 2
  Corresponding angles occupy matching positions; alternate interior angles lie between the parallel lines on opposite sides of the crossing line.
3. Hint 3
  Co-interior angles lie between the parallel lines on the same side of the crossing line and add to $180^\circ$ .
Answer

$u=71^\circ$ , corresponding; $v=71^\circ$ , alternate interior; $w=109^\circ$ , co-interior (same-side interior).

Full solution

The full marked angle at the upper crossing is
$28^\circ+43^\circ=71^\circ$

Angle $u$ has the same position as that opening, below its parallel line and to the left of the crossing line, so they are corresponding angles.
Because the lines are parallel,
$u=71^\circ$

Angle $v$ lies between the parallel lines on the opposite side of the crossing line from the given opening.
These are alternate interior angles, so
$v=71^\circ$

Angle $w$ lies between the parallel lines on the same side of the crossing line as the given opening.
These are co-interior angles.
$w=180^\circ-71^\circ$ $w=109^\circ$

At the lower crossing, $u$ and $v$ are vertical angles and agree; $v$ and $w$ form a straight angle and add to $180^\circ$ .

Answer

$u=71^\circ$ , corresponding; $v=71^\circ$ , alternate interior; $w=109^\circ$ , co-interior (same-side interior).

Key idea

When a line crosses parallel lines, the positions of the angles determine whether their measures are equal or supplementary.
Problem 6: Mark complete
Problem 7 The support drawing

For the three complete lines in the figure, list every pair guaranteed to be parallel and every pair guaranteed to be perpendicular. Classify the angles where $t$ meets $b$ .

Lines $a$ , $b$ and $t$ , with one right-angle mark.

Text description of this figure

Two horizontal lines, a above b, have arrowheads at both ends and matching parallel marks. A vertical line t, also with arrowheads at both ends, crosses both of them. Where t meets a, a small right-angle square sits in the upper-right opening. Where t meets b, nothing is marked. There are no points, lengths or measures.
1. Hint 1
  The matching marks and the small square provide different kinds of information.
2. Hint 2
  The square gives one angle where $t$ meets $a$ ; corresponding angles carry that information to line $b$ .
Answer

Parallel: $a,b$ . Perpendicular: $a,t$ and $b,t$ . All four angles where $t$ meets $b$ are right angles.

Full solution

The matching parallel marks state that $a$ and $b$ are parallel.
The square at the crossing of $a$ and $t$ marks a $90^\circ$ angle, so those two lines are perpendicular.

The corresponding angle where $t$ meets $b$ is also $90^\circ$ , because $a$ and $b$ are parallel.
Thus $t$ is perpendicular to $b$ .
The opposite angle is equal, and each neighboring angle is
$180^\circ-90^\circ=90^\circ$

So all four angles at the lower crossing are right angles.
Each of the three possible line pairs has now been accounted for.

Answer

Parallel: $a,b$ . Perpendicular: $a,t$ and $b,t$ . All four angles where $t$ meets $b$ are right angles.

Key idea

A line perpendicular to one of two parallel lines is perpendicular to the other as well.
Problem 7: Mark complete
Problem 8 The measurement record

The figure shows two straight lines crossing. A record lists three of the four openings: top $76^\circ$ , right $106^\circ$ , and bottom $74^\circ$ . Exactly one of these entries is wrong. Which entry is it, and what should it read?

Two crossing lines, with the four openings named.

Text description of this figure

Two straight lines, each with arrowheads at both ends, cross at one point to form an X. Each of the four openings at the crossing is marked with its own arc and named in words: top, right, bottom and left. The top and bottom openings are narrow, acute angles, and the left and right openings are wide, obtuse angles. No angle measures, points or equality marks are shown.
1. Hint 1
  Which pairs of openings must be equal, and which must add to $180^\circ$ ?
2. Hint 2
  Compare the top and bottom entries with each other, then check each of them against the right entry.
3. Hint 3
  Suppose each entry in turn is the wrong one, and ask whether the other two entries then agree.
Answer

The top entry; it should be $74^\circ$ .

Full solution

The top and bottom openings are opposite each other, so they are vertical angles and must be equal.
The record gives $76^\circ$ and $74^\circ$ , so the top entry or the bottom entry is wrong.

Neighboring openings form a linear pair, so each neighboring pair adds to $180^\circ$ .
The right and bottom entries pass this check, since
$106^\circ+74^\circ=180^\circ.$
The top and right entries fail it, since
$76^\circ+106^\circ=182^\circ.$

If only the right entry were wrong, the top and bottom entries would still disagree.
If only the bottom entry were wrong, the top and right entries would still fail to add to $180^\circ$ .
So the wrong entry is the top one.

The top opening must equal the bottom one, so it should read $74^\circ$ .
Checking, the top and bottom now match, and
$74^\circ+106^\circ=180^\circ.$

Answer

The top entry; it should be $74^\circ$ .

Key idea

At a crossing of two lines, every recorded measure must fit all of its pairs at once: opposite openings match, and neighboring openings add to $180^\circ$ .
Problem 8: Mark complete
Problem 9 Two angle cards

Two cards each show an acute angle. Nia says two acute angles can add to an obtuse angle. Omar says two acute angles must add to an obtuse angle. Who is correct? Give measures that support each decision.
1. Hint 1
  An obtuse angle must be greater than $90^\circ$ ; ask whether two acute angles are always large enough to pass it.
2. Hint 2
  One example shows that something can happen, and one counterexample shows that it need not happen.
3. Hint 3
  Try two acute measures that are both close to $90^\circ$ , and then two that are both small.
Answer

Nia is correct: for example, $64^\circ$ and $38^\circ$ add to an obtuse $102^\circ$ . Omar is not: for example, $22^\circ$ and $19^\circ$ add to an acute $41^\circ$ .

Full solution

Take $64^\circ$ and $38^\circ$ .
Each is between $0^\circ$ and $90^\circ$ , so each is acute.
Their sum is
$64^\circ+38^\circ=102^\circ$

Since $102^\circ$ is between $90^\circ$ and $180^\circ$ , the sum is obtuse.
So two acute angles can add to an obtuse angle, and Nia is correct.

Now take $22^\circ$ and $19^\circ$ , which are also both acute.
Their sum is
$22^\circ+19^\circ=41^\circ$

That sum is acute, not obtuse, so two acute angles need not add to an obtuse angle.
This one counterexample shows that Omar is not correct.

Answer

Nia is correct: for example, $64^\circ$ and $38^\circ$ add to an obtuse $102^\circ$ . Omar is not: for example, $22^\circ$ and $19^\circ$ add to an acute $41^\circ$ .

Key idea

The sum of two acute angles may or may not be obtuse: an example shows what can happen, and a counterexample shows what need not happen.
Problem 9: Mark complete
Problem 10 The two settings

Two machines each have two settings. At each setting, a machine makes a first angle less than $90^\circ$ and a second angle adjacent to it, sharing a vertex and a side with no overlap. On the first machine the second angle is the supplement of the first angle; on the second machine it is the complement.

Sam says that on each machine the whole opening has the same angle type at both settings, even though the first angle changes. Is Sam correct? Explain, giving the type of the whole opening on each machine at each setting.
1. Hint 1
  Consider what the words supplement and complement tell you about each pair's total.
2. Hint 2
  Call the first angle $x$ , write the whole opening on each machine in terms of $x$ , and compare each total with the angle-type definitions.
Answer

Yes. On the first machine the whole opening is straight at both settings, and on the second machine it is right at both settings.

Full solution

Let the first angle measure $x$ .
Since $x$ is less than $90^\circ$ , it has both a supplement and a complement.
The two angles are adjacent with no overlap, so the whole opening is their sum.

On the first machine, the second angle is $180^\circ-x$ , so the whole opening is
$x+(180^\circ-x)=180^\circ$
The value of $x$ cancels, so the whole opening is a straight angle at both settings.

On the second machine, the second angle is $90^\circ-x$ , so the whole opening is
$x+(90^\circ-x)=90^\circ$
Again $x$ cancels, so the whole opening is a right angle at both settings.

Changing the first angle changes its partner by the opposite amount, so Sam is correct about both machines.

Answer

Yes. On the first machine the whole opening is straight at both settings, and on the second machine it is right at both settings.

Key idea

An angle and its supplement always total $180^\circ$ , and an angle and its complement always total $90^\circ$ , so the first angle's size does not change the whole opening's type.
Problem 10: Mark complete

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