ABCD is a cyclic quadrilateral.If AB=CD,prove that angle b=angle c
To prove that angle B is equal to angle C in a cyclic quadrilateral ABCD, where AB = CD, we will use the properties of cyclic quadrilaterals.
Given: ABCD is a cyclic quadrilateral, AB = CD.
Proof:
Step 1: Draw the diagonal AC of the cyclic quadrilateral ABCD.
A _________ B
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D _________ C
Step 2: Since ABCD is a cyclic quadrilateral, the opposite angles of the quadrilateral add up to 180 degrees.
∠A + ∠C = 180° ---- (1) (Opposite angles in a cyclic quadrilateral)
Step 3: Since AB = CD (given), and opposite sides of a cyclic quadrilateral are equal, we have:
AB = CD ---- (2) (Given)
Step 4: In triangle ABC and triangle CDA, we have the following:
∠B + ∠C + ∠A = 180° ---- (3) (Sum of angles in a triangle)
∠C + ∠D + ∠A = 180° ---- (4) (Sum of angles in a triangle)
Step 5: Subtracting equation (4) from equation (3), we get:
∠B - ∠D = 0 ---- (5)
Step 6: Adding equation (5) and equation (2), we have:
∠B - ∠D + ∠C = ∠C ---- (6)
Step 7: From equation (6), we get:
∠B + ∠C = ∠C + ∠D ---- (7)
Step 8: Using equation (1), we can substitute ∠A + ∠C with 180°:
180° - ∠C + ∠C = ∠C + ∠D
Step 9: Simplifying equation (8), we have:
180° = ∠C + ∠D
Step 10: By observing equations (7) and (9), we can conclude that:
∠B = ∠D
Therefore, we have proven that angle B is equal to angle C in a cyclic quadrilateral ABCD when AB = CD.
To prove that angle B is equal to angle C in a cyclic quadrilateral ABCD, where AB = CD, we can use the property of angles subtended by the same arc.
Here's how you can prove it step by step:
Step 1: Draw a diagram of the given cyclic quadrilateral ABCD.
Step 2: Label the angles of interest. Let angle A be angle B, and let angle C be angle D.
Step 3: Use the property of a cyclic quadrilateral that opposite angles add up to 180 degrees. Therefore, angle B + angle D = 180 degrees, and angle A + angle C = 180 degrees.
Step 4: Since AB = CD, we can conclude that the arcs AB and CD are congruent.
Step 5: Use the property of a circle that equal arcs subtend equal angles. Therefore, angle B is equal to angle D, and angle A is equal to angle C.
Step 6: Write the proof using logical reasoning and the properties mentioned above.
Proof:
Given: ABCD is a cyclic quadrilateral, AB = CD
To Prove: Angle B = Angle C
Proof:
- Opposite angles in a cyclic quadrilateral are supplementary. Therefore, angle B + angle D = 180 degrees, and angle A + angle C = 180 degrees.
- AB = CD, so the arcs AB and CD are congruent.
- Equal arcs subtend equal angles on a circle. Thus, angle B is equal to angle D, and angle A is equal to angle C.
- Since angle A is equal to angle C, we can conclude that angle B is equal to angle C.
Therefore, we have proven that if a quadrilateral ABCD is cyclic and AB = CD, then angle B is equal to angle C.