Angles in the same segment questions with answers

Questions

1. Question 1Equal angles on the same arc

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ADB$\angle ADB = \angle ACB$ (∠s in same segment)$x = 29$
2. Question 2Equal angles on the same arc

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ADB$\angle ADB = \angle ACB$ (∠s in same segment)$x = 45$
3. Question 3With crossing chords

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle. The chords $AC$ and $BD$ meet at $X$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACD$\angle ACD = \angle ABD$ (∠s in same segment)$\angle ACD = 26^\circ$
   2. Step 2: Angles in triangle XCD add up to 180°$\mathstrut \angle DXC = 180^\circ - 26^\circ - 44^\circ$ (∠ sum of △)$\angle DXC = 110^\circ$
   3. Step 3: Find ∠AXB$x = 110$ (vert. opp. ∠s)
4. Question 4With crossing chords

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle. The chords $AC$ and $BD$ meet at $X$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACD$\angle ACD = \angle ABD$ (∠s in same segment)$\angle ACD = 31^\circ$
   2. Step 2: Angles in triangle XCD add up to 180°$\mathstrut \angle DXC = 180^\circ - 31^\circ - 44^\circ$ (∠ sum of △)$\angle DXC = 105^\circ$
   3. Step 3: Find ∠AXB$x = 105$ (vert. opp. ∠s)