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What is the sum of the interior angles of a triangle?
180°
270°
90°
360°
The sum of the interior angles of any triangle is always 180 degrees, a fundamental result in Euclidean geometry derived from parallel line properties. This holds true for all types of triangles, whether scalene, isosceles, or equilateral. It can be visualized by dividing the triangle into two right triangles or by using the fact that parallel lines cut by a transversal create supplementary angles. Learn more about triangle angle sums.
In triangle similarity, AAA means triangles are similar if they have...
Three equal angles
Two equal sides
Three proportional sides
One right angle
AAA stands for Angle-Angle-Angle, indicating that if all three corresponding angles of two triangles are congruent, then the triangles are similar. The side lengths of the triangles will then be proportional. This criterion does not require side length measurements. More on triangle similarity.
If two triangles are similar, which of the following must be proportional?
Corresponding sides
Corresponding angles
Altitudes only
Medians only
For two triangles to be similar, their corresponding sides must be in proportion while their corresponding angles are congruent. The proportionality of sides ensures that the shapes maintain the same overall form at different scales. This is the foundation of side-based similarity criteria like SAS and SSS. Explore side proportionality in similar triangles.
The ratio of perimeters of two similar figures is equal to...
Ratio of areas
Ratio of volumes
Difference of sides
Ratio of corresponding sides
When two figures are similar, all linear measures scale by a constant factor. This means the ratio of the perimeters is identical to the ratio of any pair of corresponding sides. Areas scale by the square of this ratio, and volumes scale by the cube. See more on similarity ratios.
An equilateral triangle has each interior angle measuring...
90°
30°
60°
45°
An equilateral triangle has three equal sides and three equal interior angles. Since the sum of angles in any triangle is 180°, each angle in an equilateral triangle is 180° ÷ 3 = 60°. This property is unique to equilateral triangles. Learn about equilateral triangles.
In a right triangle with legs 3 and 4, the hypotenuse is...
7
6
4
5
By the Pythagorean theorem, a² + b² = c², where c is the hypotenuse. For legs 3 and 4, 3² + 4² = 9 + 16 = 25, so c = ?25 = 5. This classic 3-4-5 triangle is the simplest integer right triangle. Study the Pythagorean theorem.
If the scale factor between two similar triangles is 3:4, then the ratio of their areas is...
3:4
4:9
6:8
9:16
When two figures are similar with linear scale factor k, their areas scale by k². Here k = 3/4, so the area ratio is (3/4)² = 9/16. This principle applies to all planar similar figures. More on area ratios.
Which of these is a criterion for triangle similarity (SAS)?
Three sides proportional
Two sides proportional and included angle equal
Two sides equal and included angle proportional
Two angles equal and included side proportional
The SAS similarity criterion states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. This combines side ratios with one angle congruence. Review SAS similarity.
If ?ABC ~ ?DEF and AB/DE = 2/5, then BC/EF equals...
2/3
5/2
3/5
2/5
In similar triangles, all corresponding sides are in proportion by the same scale factor. If AB/DE = 2/5, then every corresponding side ratio, including BC/EF, must also be 2/5. Consistent ratios define similarity. See corresponding side ratios.
In similar triangles, if one triangle has sides 6 and 8, the corresponding sides in the similar triangle are 9 and x. What is x?
8
10
6
12
The scale factor from the smaller to the larger triangle is 9/6 = 3/2. Multiply the second side 8 by 3/2 to get x = 8 × 1.5 = 12. Corresponding sides in similar triangles scale by the same factor. Practice similar triangle proportions.
Two squares are similar with side lengths in ratio 5:7. What is the ratio of their areas?
35:49
7:5
5:7
25:49
For any two similar figures, area scales by the square of the side ratio. Here, (5/7)² = 25/49 gives the area ratio of the smaller square to the larger. This applies to all polygons under similarity. Learn about area scaling.
Triangle ABC has angles 50° and 60°. What is the measure of the third angle?
70°
50°
90°
80°
The sum of angles in a triangle is 180°. Subtracting the known angles: 180° - (50° + 60°) = 70°. This direct application of the triangle angle sum property is fundamental. More on triangle angles.
In ?XYZ, XY = 10 and YZ = 15. ?ABC is similar with side AB = 20. What is BC?
25
20
30
15
The scale factor from ?XYZ to ?ABC is 20/10 = 2. Multiply YZ = 15 by 2 to find BC = 30. All corresponding sides are proportional by this factor. Examples of similar triangle calculations.
The ratio of the areas of two similar triangles is 16:25. What is the ratio of their corresponding sides?
8:10
16:25
5:4
4:5
If the area ratio is 16:25, then the side ratio is the square root of that, ?(16/25) = 4/5. This reverse application of the area scaling rule finds the linear scale factor. More on area and side ratios.
Which of the following is NOT a valid triangle similarity criterion?
RHS
SSS
AAS
SAS
RHS (Right angle - Hypotenuse - Side) is a congruence criterion for right triangles, not a similarity test. AAS, SAS, and SSS can all establish triangle similarity when appropriately applied. Triangle similarity vs. congruence.
In similar polygons, corresponding angles are...
supplementary
equal
proportional
complementary
By definition of similarity, corresponding angles of similar shapes are congruent, meaning they have the same measure. Only side lengths scale by a factor; angles remain equal. Understanding similarity.
A triangle has side lengths in the ratio 3:4:5. If the shortest side is 9, the longest side is...
9
12
15
18
The scale factor is 9/3 = 3. Multiply the longest-ratio value 5 by 3 to get 15. Ratios preserve proportional relationships in similar figures. Ratios in similar triangles.
If two similar triangles have perimeters of 14 and 21, the scale factor is...
2:3
14:21
3:2
21:14
Corresponding perimeters in similar figures scale by the same factor as corresponding sides. Here, 14 to 21 simplifies to 2:3, which is the linear scale factor. See how perimeters scale.
In ?ABC, a median from A meets BC at D. Which statement is true?
BD = DC
AD = BC
AD bisects ?A
AD is perpendicular to BC
A median connects a vertex to the midpoint of the opposite side, so BD = DC. Medians do not necessarily bisect angles or form right angles. More on medians in triangles.
The Angle Bisector Theorem states that the angle bisector of ?A in ?ABC divides BC into segments proportional to...
BC/AC
AB/AC
AC/AB
AB + AC
The Angle Bisector Theorem says that the bisector of ?A divides the opposite side BC into segments BD and DC such that BD/DC = AB/AC. This proportionality is a key property in triangle geometry. Angle Bisector Theorem explained.
In the coordinate plane, ?ABC with A(0,0), B(2,0), C(0,2) is similar to ?DEF. If D is at (0,0) and E is at (4,0), what are the coordinates of F?
(2,2)
(0,4)
(4,2)
(2,4)
The scale factor from ?ABC to ?DEF is 4/2 = 2. Multiplying C's coordinates (0,2) by 2 gives F = (0,4). Similarity preserves angles and proportional side lengths in the coordinate plane. Similar triangles on the coordinate plane.
Two triangles are similar, and one has area 36 while the other has area 64. A side in the first is 9. What is the corresponding side in the second?
12
8
18
16
The area ratio is 36:64 = 9:16, so the side ratio is ?(9/16) = 3/4. To go from smaller to larger, multiply by 4/3, giving 9 × (4/3) = 12. Area and side ratio relationships.
The medians of a triangle intersect at a point that divides each median in a ratio of...
2:1
1:2
3:1
1:1
The intersection point of medians is the centroid, which is located two-thirds of the way from each vertex to the midpoint of the opposite side, creating a 2:1 ratio. Learn about the centroid.
In ?ABC, AB = 8, AC = 6, BC = 10. Is it a right triangle?
No
Depends on angles
Yes
Only if acute
Check the Pythagorean theorem: 8² + 6² = 64 + 36 = 100, which equals BC² = 10². Since a² + b² = c², the triangle is right-angled at A. Review right triangle tests.
A triangle's exterior angle is equal to:
Sum of the two opposite interior angles
Half the sum of the remote interior angles
Supplementary to the adjacent interior angle
Difference of the two opposite interior angles
An exterior angle of a triangle equals the sum of its two non-adjacent interior angles. This follows from the straight-line supplementary angle concept. Exterior angle theorem.
If two similar triangles have corresponding altitudes 5 and 15, their area ratio is:
1:3
25:225
1:9
9:1
The scale factor of altitudes is 5/15 = 1/3. Area scales by the square of the linear factor: (1/3)² = 1/9. Hence the area ratio is 1:9. Altitude and area scaling.
Which point is equidistant from all vertices of a triangle?
Centroid
Incenter
Circumcenter
Orthocenter
The circumcenter is the center of the circumscribed circle and is equidistant from all triangle vertices. It is found at the intersection of the perpendicular bisectors of the sides. Discover the circumcenter.
In ?ABC, if AB/BC = 2/3 and angle B is bisected, which theorem applies?
Midpoint Theorem
Pythagorean Theorem
SSS Postulate
Angle Bisector Theorem
The Angle Bisector Theorem states that the bisector of an angle divides the opposite side into segments proportional to the adjacent sides. With AB/BC = 2/3, this theorem directly applies. Angle bisector theorem details.
In ?ABC, medians AD, BE, and CF concur at centroid G. If AG = 4, find GD.
2
6
8
4
The centroid divides each median in a 2:1 ratio, counting from the vertex to the midpoint of the opposite side. If AG = 4, then GD = 4/2 = 2. Learn about centroids.
Which point of concurrency always lies inside the triangle and is equidistant from all sides, allowing a circle to be inscribed?
Circumcenter
Orthocenter
Centroid
Incenter
The incenter is the intersection of the angle bisectors and is equidistant from all sides of the triangle, making it the center of the inscribed circle. It always lies within the triangle. About the incenter.
What are the coordinates of the centroid of ?ABC with vertices A(1,2), B(4,6), and C(7,2)?
(4, 10/3)
(4, 3)
(4, 4)
(5, 3)
The centroid's coordinates are the average of the vertices' coordinates: x = (1+4+7)/3 = 4, y = (2+6+2)/3 = 10/3. This formula applies to all triangles in the plane. Centroid in coordinate geometry.
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Study Outcomes
Understand Similar Figures -
Identify and explain the defining properties of similar figures, including corresponding angles and proportional sides, to build a strong foundation for geometry unit 5 review.
Calculate Scale Factors -
Compute and apply scale factors in various scale drawings, ensuring accurate size transformations and real-world modeling skills for your unit 5 geometry test.
Apply Triangle Similarity Criteria -
Use AA, SAS, and SSS criteria to determine triangle similarity, reinforcing your ability to recognize congruent angles and proportional sides.
Interpret and Construct Scale Drawings -
Create, analyze, and adjust scale drawings to represent geometric figures accurately, integrating practical design and measurement techniques.
Solve for Unknown Side Lengths -
Set up and solve proportions to find missing side lengths in similar figures and triangles, enhancing problem-solving skills for your geometry unit 5 test review.
Evaluate Test Performance -
Analyze quiz results to pinpoint strengths and areas for growth, allowing targeted preparation and confidence-building before taking the unit 5 geometry test.
Cheat Sheet
Understanding Similar Figures -
Review how corresponding sides of similar figures are proportional: if ΔABC~ΔDEF then AB/DE = BC/EF = AC/DF (source: Khan Academy). In a unit 5 geometry test, being fluent with these ratios ensures quick solutions under time pressure.
Applying Scale Drawings -
Master scale factors by remembering that actual length = drawing length × scale factor; e.g., a 1:50 scale means 1 cm represents 50 cm in real life (MIT OpenCourseWare). Use the mnemonic "Scale Up to See Real" to recall this in your geometry unit 5 review.
Triangle Similarity Postulates -
Familiarize yourself with AA, SAS, and SSS criteria: two equal angles (AA), two sides in proportion and included angle equal (SAS), or three proportional sides (SSS) guarantee similarity (University of Cambridge syllabus). This unit 5 review geometry tip helps streamline proofs and problem solving.
Special Right Triangle Ratios -
Remember key ratios for 45-45-90 (1:1:√2) and 30-60-90 triangles (1:√3:2) as outlined in geometry unit 5 test review materials. A quick mnemonic "45 Two, 30 Thrice" can trigger the correct ratios during your test.
Indirect Measurement Techniques -
Use similar triangles for indirect measurement; for instance, equate the ratio of an object's height to its shadow to that of a reference stick and its shadow (University of Texas). Applying this method in your unit 5 geometry test review can simplify real-world height calculations.
