1
In triangle \(\triangle PQR\), the measure of \(\angle P\) is \(58^\circ\), and the measure of \(\angle Q\) is \(71^\circ\). What is the measure of \(\angle R\)?
Lines, angles, and triangles: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
In the figure, lines r and s are parallel, and line m is a transversal. At the intersection of lines r and m, the angle labeled \(a\) is directly opposite the angle measuring \(74^\circ\). The figure is not drawn to scale.
Which statement about \(a\) is true?
3
The figure shows parallel lines r and s cut by transversal m. At the intersection of m and r, one angle is labeled \(a\) and the angle directly opposite it is labeled \(136^\circ\). The figure is not drawn to scale.
Which statement about \(a\) is true?
4
In the figure, lines m and n are parallel and are intersected by transversal t. Angle y and the angle measuring \(58^\circ\) are corresponding angles.
What is the value of \(y\)?
Note: The figure is not drawn to scale.
5
In triangle \(ABC\), \(m\angle A=47^\circ\) and \(m\angle B=68^\circ\). In triangle \(DEF\), \(m\angle D=47^\circ\) and \(m\angle E=68^\circ\).
What additional information is needed to determine whether triangle \(ABC\) is similar to triangle \(DEF\)?
6
In triangle \(RST\), points \(U\) and \(V\) lie on sides \(RS\) and \(RT\), respectively, and segment \(UV\) is drawn parallel to \(ST\). It is given that triangle \(RUV\) is similar to triangle \(RST\).
Angle \(RUV\) is a right angle, and angle \(RST\) is also a right angle. If \(m\angle RVU=38^\circ\) and \(RT=5(RV)\), what is \(m\angle RTS\)?
Note: Figure not drawn to scale.
7
In triangle \(RST\), points \(U\) and \(V\) lie on sides \(RS\) and \(RT\), respectively, and segment \(UV\) is drawn parallel to \(ST\). It is given that triangle \(RUV\) is similar to triangle \(RST\). Angles \(\angle RUV\) and \(\angle RST\) are both right angles.
If \(m\angle RVU=38^\circ\) and \(RT=4(RV)\), what is the measure of \(\angle RTS\)?
Note: Figure not drawn to scale.
8
Triangles \(\triangle JKM\) and \(\triangle TRS\) are congruent, where vertex \(J\) corresponds to vertex \(R\), vertex \(K\) corresponds to vertex \(S\), and vertex \(M\) corresponds to vertex \(T\). Angles \(J\) and \(R\) are right angles. If \(m\angle K=38^\circ\), what is \(m\angle T\)?
9
Quadrilateral \(JKLM\) is similar to quadrilateral \(RSTU\). The following vertex correspondences are given: \(J \leftrightarrow R\), \(K \leftrightarrow S\), and \(L \leftrightarrow T\). Also, side \(JK=12\) and side \(RS=18\).
If \(m\angle M=105^\circ\), what is the measure of \(\angle U\)?
10
In the figure below, triangle ABC is a right triangle with right angle at B. Point D lies on side AB, and point E lies on side BC. Segment DE is parallel to AC.
The lengths \(BD=6\), \(DA=18\), and \(DE=\sqrt{85}\).
What is the area of triangle ABC?
Note: The figure is not drawn to scale.
11
In right triangle ABC, angle B is a right angle. Point D lies on side AB, and point E lies on side AC. Segment DE is parallel to BC, so triangle ADE is a smaller right triangle inside triangle ABC.
The following lengths are given:
- \(AD=6\)
- \(DB=12\)
- \(AE=2\sqrt{13}\)
What is the area of triangle ABC?
Note: The figure is not drawn to scale.
12
In the figure, lines \(RU\) and \(ST\) intersect at point \(V\). Segment \(RS\) is parallel to segment \(UT\). The length of \(RV\) is 8, the length of \(SV\) is 6, and the length of \(VT\) is 15.
What is the total length of \(ST\)?
Note: The figure is not drawn to scale.
Answer key and explanations
1. Answer
51
The interior angles of a triangle add to \(180^\circ\). First add the two known angles: \(58+71=129\). Then subtract from \(180\): \(180-129=51\). Therefore, the measure of \(\angle R\) is \(51^\circ\).
Choice B is the sum of the given angles, and choices A and D come from incorrect subtraction.
2. Answer
\(a=74^\circ\)
The angle labeled \(a\) and the \(74^\circ\) angle are vertical angles because they are opposite each other when two lines intersect. Vertical angles are congruent, so \(a=74^\circ\).
3. Answer
\(a=136^\circ\).
The angle labeled \(a\) and the angle labeled \(136^\circ\) are opposite angles formed by two intersecting lines. Opposite angles are vertical angles, and vertical angles are congruent. Therefore, \(a=136^\circ\).
4. Answer
58
When a transversal intersects two parallel lines, corresponding angles are congruent. Since angle \(y\) corresponds to the angle measuring \(58^\circ\), the measure of angle \(y\) is also \(58^\circ\).
5. Answer
No additional information is needed
If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar by the AA similarity criterion. Since \(\angle A\) and \(\angle D\) are both \(47^\circ\), and \(\angle B\) and \(\angle E\) are both \(68^\circ\), the triangles are already guaranteed to be similar. No additional angle information is necessary because the third angles would also match automatically.
6. Answer
\(38^\circ\)
Because triangle \(RUV\) is similar to triangle \(RST\), corresponding angles are congruent. The right angles show that vertex \(U\) corresponds to vertex \(S\), so vertex \(V\) corresponds to vertex \(T\).
Therefore, angle \(RVU\) corresponds to angle \(RTS\). Since \(m\angle RVU=38^\circ\), it follows that \(m\angle RTS=38^\circ\).
The fact that \(RT=5(RV)\) is irrelevant to finding the angle measure.
7. Answer
\(38^\circ\)
Because triangle \(RUV\) is similar to triangle \(RST\), corresponding angles are congruent. The right angles show that vertex \(U\) corresponds to vertex \(S\), so vertex \(V\) corresponds to vertex \(T\).
Therefore, \(\angle RVU\) corresponds to \(\angle RTS\). Since \(m\angle RVU=38^\circ\), it follows that \(m\angle RTS=38^\circ\).
The fact that \(RT=4(RV)\) is unrelated to finding the angle measure.
8. Answer
52
Because the triangles are congruent, corresponding angles are equal. Since vertex \(K\) corresponds to vertex \(S\), \(m\angle S=38^\circ\). In triangle \(TRS\), angle \(R\) is a right angle, so \(m\angle R=90^\circ\). The angles in a triangle sum to \(180^\circ\), so:
\(m\angle T=180-90-38=52\)
Therefore, \(m\angle T=52^\circ\).
9. Answer
\(105^\circ\)
Since the quadrilaterals are similar, corresponding angles are congruent. The given correspondences are \(J \leftrightarrow R\), \(K \leftrightarrow S\), and \(L \leftrightarrow T\), so the remaining vertices must correspond as \(M \leftrightarrow U\).
Therefore, \(m\angle U=m\angle M=105^\circ\). The side lengths are irrelevant to finding the angle measure.
10. Answer
336
Since \(DE\) is parallel to \(AC\), triangles \(BDE\) and \(BAC\) are similar by AA similarity.
First find the full length of \(AB\):
\(AB=BD+DA=6+18=24\).
The similarity scale factor from triangle \(BDE\) to triangle \(BAC\) is
\(\frac{AB}{BD}=\frac{24}{6}=4\).
In the smaller right triangle, use the Pythagorean theorem to find \(BE\):
\(BE^{2}+6^{2}=(\sqrt{85})^{2}\)
\(BE^{2}+36=85\)
\(BE^{2}=49\)
\(BE=7\).
Because the triangles are similar with scale factor 4,
\(BC=4(7)=28\).
The area of triangle \(ABC\) is
\(\frac{1}{2}(24)(28)=336\).
11. Answer
108
Since \(DE\) is parallel to \(BC\), triangles \(ADE\) and \(ABC\) are similar by AA similarity.
First find the full length of side \(AB\):
\(AB=AD+DB=6+12=18\)
The scale factor from the smaller triangle to the larger triangle is
\(\frac{AB}{AD}=\frac{18}{6}=3\)
Because corresponding sides of similar triangles are proportional,
\(AC=3(AE)=3(2\sqrt{13})=6\sqrt{13}\)
Triangle \(ABC\) is a right triangle, so use the Pythagorean theorem:
\(AB^{2}+BC^{2}=AC^{2}\)
\(18^{2}+BC^{2}=(6\sqrt{13})^{2}\)
\(324+BC^{2}=468\)
\(BC^{2}=144\)
\(BC=12\)
Now find the area of triangle \(ABC\):
\(\text{Area}=\frac{1}{2}(18)(12)=108\)
12. Answer
\(21\)
Because lines \(RU\) and \(ST\) intersect at \(V\), the angles formed at \(V\) are vertical angles. Also, since \(RS\) is parallel to \(UT\), corresponding angles are congruent. Therefore, triangle \(RVS\) is similar to triangle \(UVT\) by AA similarity.
The corresponding sides are \(SV \leftrightarrow VT\) and \(RV \leftrightarrow UV\). Set up the proportion:
\(\frac{SV}{VT}=\frac{RV}{UV}\)
Substitute the known values:
\(\frac{6}{15}=\frac{8}{UV}\)
Simplify \(\frac{6}{15}=\frac{2}{5}\):
\(\frac{2}{5}=\frac{8}{UV}\)
Cross-multiply:
\(2(UV)=40\)
\(UV=20\)
Since \(ST\) consists of segments \(SV\) and \(VT\), use segment addition:
\(ST=SV+VT=6+15=21\)
The correct answer is 21.