Question

A convex quadrilateral ABCD has right angles at A, B, and C as shown. By drawing a diagonal, show that in absolute geometry, m∠D = 90. a) True b) False

Answer 1

The statement is true. In a convex quadrilateral with right angles at A, B, and C, the angle at D will also be a right angle. This can be proven by the properties of a quadrilateral and a triangle.

Step-by-step explanation:

The statement is true. In a convex quadrilateral ABCD with right angles at A, B, and C, the angle at D will also be a right angle in absolute geometry. This is due to the fact that in a quadrilateral, the sum of the interior angles is always 360 degrees. Thus, if A, B and C are right angles (each being 90 degrees), their sum would be 270 degrees. Therefore, the remaining angle, ∠D, must be 90 degrees to make the total 360 degrees.

For instance, let the diagonal be AC. Two right angled triangles are formed, namely ∆ABC and ∆ADC. Since ∠A, ∠B, and ∠C are given as 90°, according to the properties of a triangle where the sum of all angles is 180°, for ∆ABC ∠BAC = 90° and for ∆ADC, ∠ACD = 90°. Hence, m∠D is indeed 90°.

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