Mathematics
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Fluency · Pack A

8.14 Triangles

Answer each question. Show working where needed.

Bronze

  1. A triangle has angles 65° and 80°. Find the third angle.

  2. An isosceles triangle has apex angle 40°. Find each base angle.

  3. In triangle ABC, $\widehat{ABC} = 50°$ and $\widehat{BCA} = 70°$. Find $\widehat{BAC}$.

  4. Classify the triangle with sides 3, 4, 5 cm.

  5. Can a triangle have sides 4, 5, 8 cm? Justify using the triangle inequality.

  6. State the uniqueness conditions for triangles (SSS, SAS, ASA, RHS).

  7. In an equilateral triangle, what is each interior angle?

  8. A triangle has angles 90°, 35°, $x$. Find $x$.

  9. Classify by sides: a triangle has two sides of equal length. State the name.

  10. Classify by angles: a triangle has one angle of 120°. State the type.

Silver

  1. A triangle has angles $(2x + 10)°$, $(3x - 5)°$ and $90°$. Find $x$.

  2. In triangle ABC, sides $a, b, c$ correspond to vertices $A, B, C$. If $a = 6$, $b = 8$, $c = 10$, identify which is the longest, the smallest, and find the right angle (if any).

  3. A triangle has sides 7, 9, $x$ cm. The triangle inequality requires what range for $x$?

  4. A triangle is isosceles with base 8 cm and equal sides 6 cm each. Calculate (a) the perimeter, (b) the height from the apex to the base, (c) the area.

  5. Two triangles have sides 3, 4, 5 and 5, 12, 13. Are they congruent? Similar?

  6. Two triangles have sides 6, 8, 10 and 12, 16, 20. Are they similar?

  7. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. If interior angles are 40° and 60°, find the exterior angle at the third vertex.

  8. A right-angled triangle has legs 5 and 12. Find the hypotenuse.

  9. A right-angled triangle has hypotenuse 25 and one leg 7. Find the other leg.

  10. A triangle has sides labelled $a$, $b$, $c$. State which side is opposite vertex $A$ and explain the naming convention.

Gold

  1. In a figure with two parallel lines and a transversal, one angle is $(3x - 10)°$ and the corresponding angle on the other line is $(2x + 20)°$. Find $x$.

  2. Triangle ABC has $A = (0, 0)$, $B = (4, 0)$, $C = (4, 3)$. (a) Find the perimeter. (b) Classify by sides and angles.

  3. A triangle has angles $(2x)°, (3x + 10)°$ and $(x - 10)°$. Find $x$ and the three angles.

  4. An isosceles triangle has perimeter 30 cm and base 8 cm. Find the equal sides.

  5. A triangle has angles $(x + 20)°$, $(2x - 10)°$, $(3x)°$. (a) Find $x$. (b) Classify the triangle.

  6. A 3D Pythagoras-flavoured problem: a rectangular room is 4 m × 3 m × 2 m. Find the longest diagonal.

  7. Two triangles with the same sides (SSS) are always congruent. True or false? Justify.

  8. A 2D figure on a coordinate plane has vertices $A(0,0), B(6,0), C(3,4)$. Show that this is an isosceles triangle and find the height from $C$ to $AB$.

  9. A right-angled triangle has legs $a$ and $b$ and hypotenuse $c$, with $a:b = 3:4$. If the perimeter is 60 cm, find each side.

  10. Triangle ABC has vertices at $A(1,2), B(5,2), C(3,6)$. Show that ABC is isosceles and find its area.

Platinum

  1. A triangle has sides 7, 24, 25. (a) Verify it is right-angled. (b) Find the area. (c) Find the height from the right angle to the hypotenuse.

  2. Two similar triangles have sides in ratio 3:4. If the smaller has perimeter 21 cm, find the larger's perimeter. If the smaller has area 9 cm², find the larger's area.

  3. A triangle ABC has $A(0,0), B(8,0), C(4,6)$. (a) Find each side. (b) Find each angle.

  4. A triangle has angles $A$, $B$, $C$ where $A:B:C = 2:3:5$. Find the angles.

  5. Two triangles have angles (40°, 60°, 80°) and (40°, 60°, 80°). Sides of the first are 5, 6, 8. Sides of the second are 10, 12, 16. (a) Show they are similar. (b) Find the ratio of areas.

  6. A triangle has area 30 cm² and base 10 cm. Find the height. Then find the angles if the triangle is isosceles with the height bisecting the base.

  7. A right-angled triangle has hypotenuse 10 cm and legs $a$ and $b$, where $a + b = 14$. Find $a$ and $b$.

  8. Verify the triangle inequality holds: a triangle has sides $a, b, c$ with $a + b > c$, $a + c > b$, $b + c > a$. If all three are equal, can the triangle still be valid?

  9. A triangle has the angles in the ratio 1:1:2. (a) Find the angles. (b) Show that this is a right-angled isosceles triangle.

  10. Two triangles have sides 4, 5, 6 and 6, 7.5, 9. (a) Are they similar? (b) If yes, find the linear and area scale factors.