Geometry: Triangles (High School)
Classifying triangles, proving congruence, and solving for missing measures
Geometry: Triangles (High School)
Classifying triangles, proving congruence, and solving for missing measures
Geometry - Grade 9-12
- 1
In triangle ABC, angle A measures 42 degrees and angle B measures 68 degrees. Find the measure of angle C.
Use the triangle angle sum theorem.
Angle C measures 70 degrees because the angles in a triangle add to 180 degrees, and 180 - 42 - 68 = 70. - 2
A triangle has side lengths 7 cm, 10 cm, and 15 cm. Determine whether these lengths can form a triangle. Explain your reasoning.
Check all three triangle inequality statements.
Yes, the lengths can form a triangle because the sum of any two side lengths is greater than the third side: 7 + 10 > 15, 7 + 15 > 10, and 10 + 15 > 7. - 3
Classify a triangle with side lengths 9, 9, and 14 by its sides and by its angles.
The triangle is isosceles because it has two equal sides. Since 14 squared is 196 and 9 squared plus 9 squared is 162, the triangle is obtuse because the square of the longest side is greater than the sum of the squares of the other two sides. - 4
In triangle DEF, DE is congruent to DF. If angle E measures 54 degrees, find the measures of angles D and F.
The angles opposite congruent sides are congruent.
Angle F measures 54 degrees because base angles of an isosceles triangle are congruent. Angle D measures 72 degrees because 180 - 54 - 54 = 72. - 5
Two triangles have side lengths 6, 8, 10 and 9, 12, 15. Are the triangles similar? Explain.
Compare ratios of corresponding side lengths.
Yes, the triangles are similar because the corresponding side lengths are proportional. The scale factor from the first triangle to the second is 1.5 since 9/6 = 12/8 = 15/10 = 1.5. - 6
A right triangle has legs of length 12 inches and 16 inches. Find the length of the hypotenuse.
Use a squared plus b squared equals c squared.
The hypotenuse is 20 inches because 12 squared plus 16 squared equals 144 + 256 = 400, and the square root of 400 is 20. - 7
A right triangle has a hypotenuse of 13 meters and one leg of 5 meters. Find the length of the other leg.
The other leg is 12 meters because 5 squared plus b squared equals 13 squared, so 25 + b squared = 169, b squared = 144, and b = 12. - 8
Find the area of a triangle with a base of 18 cm and a height of 7 cm.
Use A = 1/2 bh.
The area is 63 square centimeters because the area of a triangle is one half times base times height, so one half times 18 times 7 equals 63. - 9
Triangle JKL is congruent to triangle MNO. The congruence statement is triangle JKL congruent to triangle MNO. If angle K measures 47 degrees, which angle in triangle MNO is congruent to angle K, and what is its measure?
Match the letters in the same positions in the two triangle names.
Angle N is congruent to angle K, so angle N measures 47 degrees. The order of the letters in the congruence statement shows the corresponding vertices. - 10
Decide whether the triangles are congruent. Triangle ABC has AB = 5, AC = 8, and angle A = 60 degrees. Triangle DEF has DE = 5, DF = 8, and angle D = 60 degrees. Explain which congruence theorem applies.
The triangles are congruent by SAS because two corresponding sides and the included angle between them are congruent. - 11
A 30-60-90 triangle has a shorter leg of 6 units. Find the longer leg and the hypotenuse.
The shorter leg is opposite the 30 degree angle.
The longer leg is 6 square root 3 units, and the hypotenuse is 12 units. In a 30-60-90 triangle, the side ratio is x to x square root 3 to 2x. - 12
A 45-45-90 triangle has legs of length 11 units each. Find the length of the hypotenuse.
The hypotenuse is 11 square root 2 units. In a 45-45-90 triangle, the hypotenuse equals a leg times square root 2. - 13
In triangle PQR, segment S connects the midpoints of sides PQ and PR. If QR = 24 cm, find the length of midsegment S.
A midsegment has half the length of the side it is parallel to.
The midsegment is 12 cm because a triangle midsegment is parallel to the third side and has half its length. - 14
The sides of a triangle are 8, 15, and 17. Determine whether the triangle is acute, right, or obtuse.
Compare the square of the longest side with the sum of the squares of the other two sides.
The triangle is right because 8 squared plus 15 squared equals 64 + 225 = 289, and 17 squared is also 289. - 15
Two sides of a triangle are 10 cm and 14 cm. What are the possible integer values for the third side length?
The possible integer values are 5 cm through 23 cm. By the triangle inequality, the third side x must be greater than 14 - 10 and less than 14 + 10, so 4 < x < 24.