Aimathic
Login | English | Deutsch

Free math worksheets

Build your own math worksheets from 30,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Angle-angle similarity applications

Click problems to add them to your worksheet.

5512358
Are the two triangles shown similar? Name the similarity criterion and cite the marked angle evidence.
Figure for problem 551235

Hints

- Focus on the angle markings rather than the apparent side lengths. - How many corresponding angle pairs are needed for the angle-angle criterion?

Solution

The marked angles show two pairs of corresponding congruent angles. Therefore, the triangles are similar by the angle-angle similarity criterion.

Answer

Yes. The triangles are similar by AA.
5521938
In the diagram, the marked angles at \(C\) and \(E\) are congruent. Which similarity criterion proves that triangle \(ABC\) is similar to triangle \(ADE\)?
Figure for problem 552193

Hints

- One pair of congruent angles is already marked. - Look for an angle that belongs to both triangles.

Solution

1. The diagram gives \(\angle C\cong\angle E\). 2. The triangles also share the angle at \(A\), because \(B\) and \(D\) lie on the same ray from \(A\), and \(C\) and \(E\) lie on the other ray. 3. Therefore, \(\triangle ABC\sim\triangle ADE\) by AA similarity.

Answer

AA similarity.
5512368
In the diagram, \(\overline{DE}\parallel\overline{BC}\). Explain why triangle \(ADE\) is similar to triangle \(ABC\) using AA. Name the two angle relationships you use.
Figure for problem 551236

Hints

- Use the stated parallel segments rather than estimating angles from the drawing. - Treat \(AB\) and \(AC\) as transversals of the parallel segments. - Identify one corresponding-angle pair on each side of the triangle.

Solution

Because \(DE\parallel BC\), corresponding angles give \(\angle ADE\cong\angle ABC\) and \(\angle AED\cong\angle ACB\). Therefore, triangle \(ADE\sim\triangle ABC\) by AA.

Answer

\(\angle ADE\cong\angle ABC\) and \(\angle AED\cong\angle ACB\), so \(\triangle ADE\sim\triangle ABC\) by AA.
5512378
The marked triangles are similar by AA. Find \(x\).
Figure for problem 551237

Hints

- Use the angle markings to match corresponding vertices before using side lengths. - Find the scale factor from a pair of corresponding sides whose lengths are known. - Apply the same scale factor to the side corresponding to \(8\).

Solution

The angle markings match \(A\) with \(D\) and \(B\) with \(E\), so \(AB\) corresponds to \(DE\) and \(AC\) corresponds to \(DF\). The scale factor from triangle \(ABC\) to triangle \(DEF\) is \(\frac{9}{6}=\frac{3}{2}\). Therefore, \(x=8\cdot\frac{3}{2}=12\).

Answer

\(x=12\)
5521948
The marked angles show that triangle \(ABC\) is similar to the other triangle. Write the correct similarity statement starting with \(ABC\), then find the scale factor from triangle \(ABC\) to its image.
Figure for problem 552194

Hints

- Match vertices by their angle markings before comparing side lengths. - Keep the vertex order consistent in the similarity statement. - Use one known pair of corresponding sides for the scale factor.

Solution

1. The one-arc angles give \(A\leftrightarrow Q\), and the two-arc angles give \(B\leftrightarrow R\). Therefore, \(C\leftrightarrow P\). 2. The correct statement is \(\triangle ABC\sim\triangle QRP\). 3. Corresponding sides \(AB=4\) and \(QR=6\), so the scale factor is \(\frac{6}{4}=\frac{3}{2}\).

Answer

\(\triangle ABC\sim\triangle QRP\), with scale factor \(\frac{3}{2}\).
5546178
The two triangles are shown with some angle measures. a) Use the triangle angle sum to find \(\angle C\) in triangle \(ABC\). b) Explain why \(\triangle ABC\) and \(\triangle DEF\) are similar by AA. c) Write the correspondence in a similarity statement.
Figure for problem 554617

Hints

- What must the three interior angles of triangle \(ABC\) add to? - After finding the missing angle, compare it with the labeled angles in the second triangle. - Match vertices by equal angle measures before writing the similarity statement.

Solution

1. In \(\triangle ABC\), the angles sum to \(180^\circ\), so \(\angle C=180^\circ-48^\circ-67^\circ=65^\circ\). 2. The diagram gives \(\angle A=48^\circ\) and \(\angle D=48^\circ\), so those angles are congruent. 3. The diagram gives \(\angle F=65^\circ\), and step 1 shows \(\angle C=65^\circ\), so a second pair of angles is congruent. 4. Therefore, the triangles are similar by AA, with \(A\leftrightarrow D\), \(B\leftrightarrow E\), and \(C\leftrightarrow F\).

Answer

a) \(\angle C=65^\circ\) b) \(\angle A=\angle D=48^\circ\) and \(\angle C=\angle F=65^\circ\), so the triangles are similar by AA. c) \(\triangle ABC\sim\triangle DEF\)
5512388
A claim says, “If two triangles each have a \(60^\circ\) angle, then they are similar by AA.” Is the statement correct? Explain, and give two possible sets of angle measures that show why.

Hints

- Recall exactly how many corresponding angle pairs the AA criterion requires. - Use the fact that the three angles in a triangle sum to \(180^\circ\). - Try constructing two different triangles that share one angle but not a second angle.

Solution

1. The statement is not correct. AA requires two pairs of corresponding congruent angles, not just one pair. 2. For example, one triangle could have angles \(60^\circ,50^\circ,70^\circ\), while another could have angles \(60^\circ,40^\circ,80^\circ\). Both contain a \(60^\circ\) angle, but the other angles do not match, so the triangles are not similar.

Answer

No. One equal angle is not enough for AA. For example, \((60^\circ,50^\circ,70^\circ)\) and \((60^\circ,40^\circ,80^\circ)\) are not similar triangles.
5512398
At the same time of day, a \(6\,\text{ft}\) person casts a \(4\,\text{ft}\) shadow, and a flagpole casts an \(18\,\text{ft}\) shadow. Assuming level ground and parallel sunlight, find the flagpole's height. Explain why the two height-shadow triangles are similar.

Hints

- Identify the right angle in each height-shadow triangle. - Parallel sun rays create another pair of equal angles. - Match height with height and shadow length with shadow length. - Set up one proportion using corresponding sides.

Solution

The person and flagpole are both vertical, so each height-shadow triangle has a right angle. Parallel sunlight creates the same acute angle with the ground in both triangles. Therefore, the triangles are similar by AA. Using corresponding height-to-shadow ratios, \(\frac{6}{4}=\frac{h}{18}\), so \(h=27\,\text{ft}\).

Answer

The flagpole is \(27\,\text{ft}\) tall.
5512418
In the diagram, \(\overline{DE}\parallel\overline{BC}\). a) Name two angle congruences that establish \(\triangle ADE\sim\triangle ABC\) by AA, and explain how the parallel lines give those congruences. b) After establishing AA similarity, find \(EC\) and \(BC\).
Figure for problem 551241

Hints

- Before using any side proportions, identify what the parallel segments imply about angles made by \(AB\) and \(AC\). - Match the vertices only after you have two angle pairs. - Then combine the two pieces on \(\overline{AB}\) and determine the scale factor.

Solution

1. Since \(DE\parallel BC\) and \(AB\) is a transversal, \(\angle ADE\cong\angle ABC\). Since \(DE\parallel BC\) and \(AC\) is a transversal, \(\angle AED\cong\angle ACB\). 2. Therefore, \(\triangle ADE\sim\triangle ABC\) by AA. 3. From the diagram, \(AD=6\) and \(DB=3\), so \(AB=9\). The scale factor from \(\triangle ADE\) to \(\triangle ABC\) is \(\frac{AB}{AD}=\frac{9}{6}=\frac{3}{2}\). 4. Thus \(AC=8\cdot\frac{3}{2}=12\), so \(EC=12-8=4\). 5. Also, \(BC=10\cdot\frac{3}{2}=15\).

Answer

a) \(\angle ADE\cong\angle ABC\) and \(\angle AED\cong\angle ACB\), from corresponding angles formed by the parallel lines and the two transversals. b) \(EC=4\) and \(BC=15\).
5521958
Points \(A,O,C\) are collinear, and points \(B,O,D\) are collinear. The marked angles at \(A\) and \(C\) are congruent. a) Explain why \(\triangle AOB\sim\triangle COD\). b) Use the labeled lengths to find \(x=OD\).
Figure for problem 552195

Hints

- Look at the pair of angles formed at the intersection \(O\). - After establishing similarity, identify which side corresponds to \(AO\). - Use the same scale factor on the side corresponding to \(OB\).

Solution

1. The diagram gives \(\angle A\cong\angle C\). 2. Angles \(\angle AOB\) and \(\angle COD\) are vertical angles, so they are congruent. 3. Therefore, \(\triangle AOB\sim\triangle COD\) by AA. 4. The scale factor from \(\triangle AOB\) to \(\triangle COD\) is \(\frac{CO}{AO}=\frac{4}{6}=\frac{2}{3}\). 5. Since \(OB=9\) corresponds to \(OD=x\), \(x=\frac{2}{3}\cdot9=6\).

Answer

a) The triangles are similar by AA. b) \(x=6\)
5546188
In the first diagram, side \(AB\) is extended past \(B\), forming the labeled exterior angle. The second diagram shows triangle \(DEF\). Explain why \(\triangle ABC\) and \(\triangle DEF\) are similar by AA. Your reasoning must use the exterior angle in the first diagram.
Figure for problem 554618

Hints

- The exterior angle and the adjacent interior angle at \(B\) form a straight angle. - After finding the interior angle at \(B\), compare angle measures across the two triangles. - Two matching angle pairs are enough to establish AA similarity.

Solution

1. The exterior angle at \(B\) is \(118^\circ\), so the adjacent interior angle is \(\angle ABC=180^\circ-118^\circ=62^\circ\). 2. The second triangle has \(\angle E=62^\circ\), so \(\angle ABC\cong\angle E\). 3. The diagrams also give \(\angle A=47^\circ\) and \(\angle D=47^\circ\). 4. Therefore, \(\triangle ABC\sim\triangle DEF\) by AA, with \(A\leftrightarrow D\), \(B\leftrightarrow E\), and \(C\leftrightarrow F\).

Answer

The interior angle at \(B\) is \(62^\circ\) because it forms a linear pair with the \(118^\circ\) exterior angle. Since \(\angle B=\angle E=62^\circ\) and \(\angle A=\angle D=47^\circ\), \(\triangle ABC\sim\triangle DEF\) by AA.
5512408
Triangles \(ABC\) and \(DEF\) satisfy \(\angle A=\angle D=50^\circ\). Also, \(\angle B=(2x+10)^\circ\) and \(\angle E=3x^\circ\). For which positive values of \(x\) are the triangles guaranteed to be similar by AA? Consider all possible correspondences of the remaining angles.

Hints

- One angle pair is already known to correspond. - Express each triangle's third angle using the \(180^\circ\) angle sum. - The second matching angle does not have to use the same named position in both triangles. - Check every possible correspondence of the two remaining angles.

Solution

Since \(\angle A=\angle D=50^\circ\), one angle pair already matches. Triangle \(ABC\) has \(\angle C=180^\circ-50^\circ-(2x+10)^\circ=(120-2x)^\circ\). Triangle \(DEF\) has \(\angle F=180^\circ-50^\circ-3x^\circ=(130-3x)^\circ\). One possibility is \(\angle B=\angle E\): \(2x+10=3x\), giving \(x=10\). The other possible correspondence is \(\angle B=\angle F\): \(2x+10=130-3x\), giving \(x=24\). Both values produce valid positive triangle angles, so both guarantee AA similarity, with different vertex correspondences.

Answer

\(x=10\) or \(x=24\)
5546198
In triangle \(ABC\), points \(D\) and \(E\) lie on \(\overline{AB}\) and \(\overline{AC}\), and \(\overline{DE}\parallel\overline{BC}\), as shown. a) Explain why \(\triangle ADE\sim\triangle ABC\) using angle relationships created by the parallel lines. b) Find \(BC\).
Figure for problem 554619

Hints

- Treat \(AB\) and \(AC\) as transversals crossing the parallel segments. - Identify which angles in the small triangle correspond to angles in the large triangle before using side lengths. - Combine the two labeled pieces of \(AB\) before forming a similarity ratio.

Solution

1. Because \(DE\parallel BC\), transversal \(AB\) gives \(\angle ADE\cong\angle ABC\), and transversal \(AC\) gives \(\angle AED\cong\angle ACB\). 2. Therefore, \(\triangle ADE\sim\triangle ABC\) by AA. 3. From the diagram, \(AD=4\) and \(DB=2\), so \(AB=6\). 4. The small-to-large scale factor is \(\frac{AD}{AB}=\frac{4}{6}=\frac{2}{3}\). 5. Since \(DE=6\) corresponds to \(BC\), \(\frac{6}{BC}=\frac{2}{3}\), so \(BC=9\).

Answer

a) \(\angle ADE\cong\angle ABC\) and \(\angle AED\cong\angle ACB\) because \(DE\parallel BC\), so the triangles are similar by AA. b) \(BC=9\)

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets