Angles in triangles questions with answers

Questions

1. Question 1Angle sum of a triangle

   The diagram shows triangle $XYZ$. Find $x$.

   Show the worked solution

   1. Step 1: Add up the three angles of the triangle$x^\circ + 53^\circ + 64^\circ = 180^\circ$
   2. Step 2: Find x$x = 180 - 53 - 64$$x = 63$ (∠ sum of △)
2. Question 2Isosceles triangles

   In triangle $XYZ$, $XY = XZ$. Find $x$.

   Show the worked solution

   1. Step 1: Use XY = XZ: which angle equals x°?$\angle XZY = x^\circ$ (base ∠s of isos. △)
   2. Step 2: Add up the three angles of the triangle$x^\circ + x^\circ + 44^\circ = 180^\circ$$2x = 136$$x = 68$ (∠ sum of △)
3. Question 3Equilateral triangles

   $ABC$ is an equilateral triangle and $D$ lies on $BC$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ABC and give the reason$\angle ABC = 60^\circ$ (equil. △)
   2. Step 2: Which angle fact gives x?$x^\circ = 60^\circ + 14^\circ$ (ext. ∠ of △)
   3. Step 3: Find x$x = 74$
4. Question 4Exterior angle of a triangle

   In the diagram, $MN$ is produced to $P$. Find $x$.

   Show the worked solution

   1. Step 1: Which angle fact gives x?$x^\circ = 77^\circ + 58^\circ$ (ext. ∠ of △)
   2. Step 2: Find x$x = 135$
5. Question 5Angles as expressions

   In the diagram, $YZ$ is produced to $W$. Find $x$.

   Show the worked solution

   1. Step 1: Write an equation in x$(x + 20)^\circ + (2x + 15)^\circ = 125^\circ$ (ext. ∠ of △)
   2. Step 2: Collect like terms and solve$3x + 35 = 125$$3x = 90$$x = 30$
6. Question 6Is the triangle isosceles?

   In the diagram, $MN$ is produced to $K$. Is triangle $LMN$ isosceles? Choose the answer with its reason.

   Show the worked solution

   1. Step 1: Write an equation in x$(x + 20)^\circ + (4x - 1)^\circ = (4x + 37)^\circ$ (ext. ∠ of △)
   2. Step 2: Collect like terms and solve$5x - 4x = 18$$x = 18$
   3. Step 3: Work out the three angles of the triangle$\angle L = 38^\circ$, $\angle M = 71^\circ$, $\angle LNM = 180^\circ - 109^\circ$$\angle LNM = 71^\circ$
   4. Step 4: Are two of the angles equal? Equal angles face equal sides$\angle M = \angle N$, so triangle $LMN$ is isosceles with $LM = LN$ (sides opposite equal angles).