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Quizzes > High School Quizzes > Mathematics

Practice Quiz 6-2: Proving Triangles Are Similar

Conquer Unit 6: Similar Triangles and Figures

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art depicting a Similar Triangles Challenge quiz for high school geometry students.

Which criterion proves triangles are similar by showing that two corresponding angles are congruent?
AA Criterion
SSS Criterion
HL Criterion
SAS Criterion
Triangles are similar if two pairs of corresponding angles are congruent because the third angle will automatically be congruent. The AA (Angle-Angle) criterion is sufficient to establish triangle similarity.
If two triangles are similar, which property always holds for their corresponding sides?
They have equal lengths
They are congruent
They are proportional
They are parallel
In similar triangles, the corresponding sides are proportional, not necessarily equal. This proportionality is central to calculating unknown side lengths in similar figures.
Which statement is always true for similar triangles?
They have the same perimeter
Corresponding angles are congruent
They have the same area
Corresponding sides are equal
Similar triangles have all corresponding angles congruent, which preserves the shape of the triangles. However, their sides are only proportional, which means they are scaled versions of one another.
Which feature in a diagram often indicates that triangles might be similar?
Equal side lengths
Presence of parallel lines
Shared vertices
Congruent medians
Parallel lines in a diagram often create congruent corresponding angles when intersecting other lines. This is a strong indicator that the triangles involved are similar due to corresponding angles being equal.
If triangle ABC is similar to triangle DEF, which of these statements is true?
Angle B is congruent to Angle D
Side AC is equal to side DF
Angle A is congruent to Angle D
Side AB is equal to side DE
In similar triangles, the correspondence between vertices implies that angle A corresponds to angle D. This congruence of corresponding angles is a direct consequence of the AA similarity criterion.
If two triangles have corresponding side lengths in the ratio 3:5, what is the scale factor from the smaller triangle to the larger triangle?
8/3
5/3
3/5
None of the above
The scale factor is found by dividing the larger side by the smaller side, which is 5 divided by 3. This ratio is applied to each corresponding side in the similar triangles.
Which set of ratios correctly demonstrates SSS similarity between two triangles ABC and DEF?
AB/DE = BC/FD = AC/EF
AB/DE = BC/EF = AC/DF
AB/FD = BC/EF = AC/DE
AB/EF = BC/DF = AC/DE
SSS similarity requires that all three pairs of corresponding sides have the same ratio. Option A correctly matches the sides of triangle ABC with the corresponding sides of triangle DEF.
In triangle ABC, if angle A = 50° and angle B = 60°, what is the measure of angle C?
70°
90°
80°
100°
The sum of the angles in a triangle is always 180°. Subtracting the measures of angle A and angle B (50° + 60°) from 180° gives angle C as 70°.
Which similarity criterion is applicable when one pair of corresponding angles is congruent and the ratio of the included sides is known?
SSS
SAS
HL
AA
The SAS similarity criterion applies when two sides in one triangle are in proportion to two sides in another triangle and the included angles are equal. This ensures the triangles have the same shape.
In a pair of similar triangles, if a side of the smaller triangle measures 4 and the corresponding side of the larger triangle measures 6, what is the simplified ratio of similarity?
6:4
4:6
2:3
3:2
The ratio of the corresponding sides is 4:6, which simplifies to 2:3. This proportion shows the relationship between the sizes of the similar triangles.
How does the intercept theorem (basic proportionality theorem) relate to similar triangles?
It confirms that the sum of the angles in a triangle is 180°.
It shows that the medians of triangles are congruent.
It states that a line parallel to one side of a triangle divides the other sides proportionally.
It establishes that triangles with equal perimeters are similar.
The intercept theorem tells us that if a line is drawn parallel to one side of a triangle, it cuts the other two sides in proportional segments. This is a fundamental concept in proving similarity between triangles.
What is a necessary step in proving two triangles are similar using the AA criterion?
Verify that the triangles are congruent.
Establish one pair of corresponding angles is equal and determine the remaining angles.
Demonstrate that the triangles have the same area.
Show that all sides are in proportion.
By establishing that one pair of corresponding angles is equal, the remaining angles must also be equal because the sum of angles in a triangle is always 180°. This satisfies the AA criterion for proving triangle similarity.
In coordinate geometry, which transformation will generate a similar triangle?
Translation
Dilation
Reflection
Rotation
Dilation scales a figure by a constant factor while preserving its shape and angles, resulting in a similar figure. This transformation is key in creating similar triangles in coordinate geometry.
If triangles ABC and DEF are similar with perimeters 24 and 36 respectively, what is the scale factor from triangle ABC to triangle DEF?
4/3
3/2
1
2/3
The scale factor is determined by dividing the perimeter of the larger triangle (36) by the perimeter of the smaller triangle (24), which simplifies to 3/2. This ratio applies to every corresponding linear measurement in the triangles.
Which method is most straightforward to solve for an unknown side in a pair of similar triangles?
Applying the Pythagorean theorem
Using angle bisectors
Assuming the unknown side is congruent to a known side
Setting up and solving a proportion between corresponding sides
Since the corresponding sides of similar triangles are proportional, setting up a proportion is the most efficient method to solve for an unknown side. This method directly utilizes the definition of similarity.
Triangle PQR ~ Triangle XYZ. If PQ = 5, PR = 7, angle P = 45°, and XY = 10, what is the length of side XZ corresponding to PR using SAS similarity?
12
14
10
7
The scale factor from triangle PQR to triangle XYZ is found by comparing PQ and XY, which gives 10/5 = 2. Multiplying PR (7) by the scale factor yields XZ = 7 × 2 = 14.
Given triangles ABC ~ DEF with AB = 8, BC = 10, CA = 6 and DE = 12, what is the corresponding length EF?
12
10
15
18
The scale factor from triangle ABC to triangle DEF is determined by AB and DE, so 12 ÷ 8 = 1.5. Applying this factor to side BC gives EF = 10 × 1.5 = 15.
In triangle PQR, a segment LM is drawn through points L on PR and M on PQ such that LM is parallel to side QR. Which smaller triangle within triangle PQR is similar to it?
Triangle LQR
Triangle PLM
Triangle LMQ
Triangle QLM
Drawing a line segment parallel to one side of a triangle creates a smaller triangle that is similar to the original triangle. Here, triangle PLM is the one formed by the intersection of LM with sides PR and PQ.
Two right triangles sharing an acute angle are compared. Which property confirms their similarity?
They have one pair of corresponding acute angles congruent, ensuring all angles are congruent
They are both inscribed in the same circle
They share the same hypotenuse
They both have legs of equal length
In right triangles, if one of the acute angles is congruent, then the other must also be congruent because the two acute angles add up to 90°. This satisfies the AA criterion, proving the triangles are similar.
In triangle ABC, point D lies on side BC and AD is an angle bisector. Given AB = 9 and AC = 12, which proportional relationship must hold according to the Angle Bisector Theorem?
AB/BD = AC/DC
AB/AC = BD/DC
AD/AB = AD/AC
AD/AC = AB/BD
The Angle Bisector Theorem states that an angle bisector divides the opposite side into segments proportional to the adjacent sides. Therefore, the correct relationship is AB/AC = BD/DC.
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Study Outcomes

  1. Identify the criteria that establish triangle similarity.
  2. Apply AA, SAS, and SSS postulates to prove triangles are similar.
  3. Calculate unknown side lengths using proportional reasoning.
  4. Analyze geometric diagrams to determine corresponding angles and sides.
  5. Evaluate problem setups and justify triangle similarity through logical arguments.

Unit 6 Similar Triangles Quiz & Test Cheat Sheet

  1. Definition of Similar Triangles - Triangles are geometry twins when their corresponding angles match and their sides stay in the same proportion. This concept is your foundation for spotting similarity in any shape, from simple homework to tricky contest problems. How To Find if Triangles are Similar
  2. Angle-Angle (AA) Criterion - If two angles of one triangle equal two angles of another, you've got a match! This quick check is like a secret handshake for triangles - no need to measure every side. Criteria for Similarity of Triangles
  3. Side-Angle-Side (SAS) Criterion - When two sides of one triangle are proportional to two sides of another and the included angle is equal, similarity is confirmed. Think of it as a proportional side party with the angle as the VIP guest. Criteria for Similarity of Triangles
  4. Side-Side-Side (SSS) Criterion - Got three proportional pairs of sides? Then your triangles are kindred spirits. This is the ultimate all-sides check for similarity. Criteria for Similarity of Triangles
  5. All Equilateral Triangles Are Similar - Every equilateral triangle has equal angles and sides, so they're all cut from the same geometric cloth. No extra work needed - these shapes are similarity superstars. Similar Triangles
  6. Basic Proportionality Theorem (Thales' Theorem) - A line parallel to one side of a triangle splits the other two sides proportionally. It's like a geometry cheat code for dividing segments. Similar Triangles | GeeksforGeeks
  7. Real-World Applications - Use similar triangles to find heights of trees, buildings, or skyscrapers by comparing shadow lengths. It's practical math that turns you into a real-life detective! Similar Triangles
  8. Area Ratio and Side Ratio - The ratio of two similar triangles' areas equals the square of their side ratios. This powerful shortcut saves time when comparing sizes on exams. Similar Triangles
  9. Spotting Similar Triangles in Diagrams - Look for parallel lines, transversals, and repeated angle patterns to spot hidden similar triangles. Practice on varied figures to sharpen your eye. How To Find if Triangles are Similar
  10. Mnemonic Magic: AA, SAS, SSS - Remember "AA, SAS, SSS" to recall your criteria instantly under exam pressure. This friendly acronym turns stress into confidence! Similar Triangle Theorems
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