Triangles

\(\text{In the triangle shown, }PQ=QR\text{. What is the value }x?\)

\(\text{In the right triangle shown, what is the value of }a?\)

\(\begin{aligned}
&\text{In triangle }ABC\text{, the measure of angle }B\mathrm{~is~}52°\text{ and the measure of angle }C\text{ is }17°.\\
&\text{What is the measure of angle }A ?\end{aligned}\)

\(\begin{aligned}&\mathrm{In~}\triangle JKL\text{, the measures of both }\angle J\mathrm{~and~}\angle K\text{ are equal and the measure of }\angle L\mathrm{~is~}136°.\\&\text{What is the measure of }\angle J?\end{aligned}\)

\(\mathrm{In~}\triangle FGH\text{, the measures of }\angle F\mathrm{~and~}\angle G\text{ are each }39°\text{. What is the measure of }\angle H?\)

\(\begin{aligned}
&\mathrm{In~}\triangle RST\text{, the measure of }\angle R\mathrm{~is~}63°.\text{ Which of the following could be the measure,}\\
&\text{in degrees, of }\angle S?\end{aligned}\)

\(\begin{aligned}&\text{In a triangle }ABC\text{, the measure of angle }A\mathrm{~is~}48°\text{, the measure of angle }B\mathrm{~is~}90°,\\&\text{and the measure of angle }C\mathrm{~is~}\left(\frac{k}{2}\right)^{\circ}\text{. What is the value of }k?\end{aligned}\)

\(\begin{aligned}&\text{In the triangle above, }a=34\text{. What is the value of }b+c?\end{aligned}\)

\(\text{In the figure above, }\overline{AD}\text{ intersects }\overline{BE}\text{ at C. If }x=100,\text{what is the value of }y?\)

\(\begin{aligned}
&\mathrm{In~the~figure~above},\overline{MQ}\mathrm{~and~}\overline{NR}\text{ intersect at point~}P,NP=QP,\mathrm{and~}MP=PR.\\
&\text{What is the measure, in degrees, of }\angle QMR?\end{aligned}\)

\(\begin{aligned}&\text{In triangle }PQR,\overline{QR}\text{ is extended to point }S.\text{ The measure of }\angle PQR\mathrm{~is~}142°\text{, and the}\\&\text{measure of }\angle PRS\mathrm{~is~}166°\text{. What is the measure of }\angle QPR?\end{aligned}\)

\(\begin{aligned}&\text{In triangle }PQR\text{, the measure of angle }P\mathrm{~is~}(3x+5)°\text{, the measure of angle }Q\mathrm{~is~}(2x+9)°,\\
&\text{and the measure of angle }R\mathrm{~is~}(4y+5)°\text{. If side }QR\text{ is extended through point }R\text{ to point }S,\\&\text{and the measure of angle }PRS\mathrm{~is~}(x+y)°\text{, what is the value of }x+y?\end{aligned}\)

\(\text{In the figure above, }RT=TU\text{. What is the value of }x?\)

\(\begin{aligned}
&\text{In the figure, }RT=TU\text{, the measure of angle }VST\mathrm{~is~}23°\text{, and the}\\&\text{measure of angle }RVS\mathrm{~is~}34°\text{. What is the value of }x?\end{aligned}\)

\(\begin{aligned}&\text{In the figure shown, the length of }\overline{AB}\text{ is 66. Which statement provides sufficient information}\\&\text{to prove that triangle }ABC\text{ is isosceles?}\end{aligned}\)

\(\begin{aligned}
&\text{For isosceles triangle }ABC,AB=BC=41,\text{ the measure of angle }ABC\text{ is 36°, point }D\mathrm{~lies}\\
&\mathrm{on~}\overline{BC},\mathrm{~and~}\angle BAC\text{ is bisected by }\overline{AD}.\text{ Which of the following statements must be true?}\end{aligned}\)

\(\begin{aligned}
&\mathrm{In~the~figure,~parallel~lines~}a\mathrm{~and~}b\text{ are intersected by lines }c,d,\mathrm{and~}e.\mathrm{~If~}z=49,y=136,\\&\mathrm{and~}v<z,\text{which statement about }x\mathrm{~and~}w\mathrm{~must~be~true?}\end{aligned}\)

\(\begin{aligned}&\text{In the figure, parallel lines }r\text{ and }s\text{ are intersected by lines }t,u,\mathrm{~and~}w.\mathrm{~If~}d=42,g=143,\\&\mathrm{and~}h>d,\text{which statement about }j\mathrm{~and~}p\mathrm{~must~be~true?}\end{aligned}\)

\(\begin{aligned}&\text{The triangle inequality theorem states that the sum of any two sides of a triangle must be}\\&\text{greater than the length of the third side, If a triangle has side lengths of 8 and 13, which}\\&\text{inequality represents the possible lengths, }x\text{, of the third side of the triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The length of two sides of a triangle are 4 centimeters and 6 centimeters. If the perimeter of}\\
&\text{the triangle is 18 centimeters, what is the length, in centimeters, of the third side of this triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The perimeter of an isosceles triangle 83 inches. Each of the two congruent sides of the}\\&\text{triangle has a length of 24 inches. What is the length, in inches, of the third side?}\end{aligned}\)

\(\begin{aligned}
&\text{The figure shows the lengths, in inches, of two sides of a right triangle. What is the area of}\\
&\text{the triangle, in square inches?}\end{aligned} \)

\(\begin{aligned}
&\text{A triangle has a base length of 82 centimeters and a height of 164 centimeters. What is}\\
&\text{the area, in square centimeters, of the triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The area of triangle }ABC\text{ is 220 square centimeters. What is the height }h\text{, in centimeters of}\\
&\text{this triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The area of a triangle is equal to }x^{2}\text{ square centimeters. The length of the base of the}\\&\text{triangle is }2x+6\text{ centimeters, and the height of the triangle is }x-2\text{ centimeters. }\\&\text{What is the values of }x?\end{aligned}\)

\(\begin{aligned}
&\text{The area of a triangle is 270 square centimeters. The length of the base of the triangle is }\\
&\text{12 centimeters greater than the height of the triangle. What is the height, in centimeters, }\\&\text{of the triangle?}\\\end{aligned}\)

\(\begin{aligned}&\text{The area of a triangle is 224 square inches. The length of the base of the triangle, in inches, is}\\&\text{the sum of 15 and a positive number }d\text{, and the height, in inches, is 3 less than }d\text{. What is the}\\&\text{value of }d?\end{aligned}\)

\(\begin{aligned}
&\text{A right triangle has sides of length }2\sqrt{2},6\sqrt{2}\text{, and}\sqrt{80}\text{ units. What is the area of}\\
&\text{the triangle, in square units?}\end{aligned}\)

\(\begin{aligned}&\text{The line segment shown in the }xy\text{-plane represents one of the legs of a right triangle. The}\\&\text{area of this triangle is }48\sqrt{13}\text{ square units. What is the length, in units, of the other leg of}\\&\mathrm{this~triangle?}\end{aligned}\)

\(\text{For the right triangle shown, }a=4\text{ and }b=5\text{. Which expression represents the value of c?}\)

\(\text{In the right triangle shown, what is the value of }x?\)

\(\begin{aligned}&\text{In triangle }ABC,\mathrm{angle~}B\text{ is a right angle. The length of side }AB\mathrm{~is~}10\sqrt{39}\text{ and the length of}\\&\mathrm{side~}BC\mathrm{~is~}24\sqrt{39}.\text{ What is the length of side }AC?\end{aligned}\)

\(\begin{aligned}
&\text{One leg of a right triangle has a length of 43.2 millimeters. The hypotenuse of the triangle has}\\
&\text{a length of 196.8 millimeters. What is the length of the other leg of the triangle, in millimeters?}\end{aligned}\)

\(\begin{aligned}
&\text{A right triangle has legs with lengths of 24 centimeters and 21 centimeters. If the length of this}\\
&\text{triangle’s hypotenuse, in centimeters, can be written in the form 3}\sqrt{d},\mathrm{where~}d\mathrm{~is~an~integer,}\\&\text{what is the value of }d ?\end{aligned}\)

\(\text{In the triangle shown, what is the value of }x?\)

\(\begin{aligned}
&\text{An equilateral triangle has a height of 47}\sqrt{3}\text{ units. What is the length, in units, of one side of}\\
&\text{this triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is }k\sqrt{3}\\
&\text{centimeters, where }k\text{ is a constant. What is the value of }k?\end{aligned}\)

\(\begin{aligned}
&\text{An isosceles right triangle has a hypotenuse of length 58 inches. What is the perimeter,}\\
&\text{in inches, of this triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{An isosceles right triangle has a perimeter of }94+94\sqrt{2}\text{ inches. What is the length, in inches, }\\
&\text{of one leg of this triangle?}\end{aligned}\)

\(\begin{aligned}
&\text{The perimeter of an isosceles triangle is }38+38\sqrt{2}\text{ inches. What is the length, in inches,}\\
&\text{of the hypotenuse of this triangle?}\end{aligned}\)