Question

Suppose ABCD is a rhombus and that the bisector of ∠ABD meets

AD
at point K. Prove that m∠AKB = 3m∠ABK.
m∠AKB = m∠KBD + m∠
by reason
Find the angle that missing angle so that angle kbd and that angle will equal angle akb.
explain

Answer 1

• missing angle: ∠DBC

Explanation:

Proof:

• m∠ABK ≅ m∠KBD — given that BK bisects ∠ABD
• m∠ABD = m∠ABK + m∠KBD = 2·m∠ABK
• m∠ABD ≅ m∠DBC — properties of a rhombus: a diagonal bisects the angles
• m∠DBC = 2·m∠ABK — transitive property (both equal to m∠ABD)
• m∠KBC = m∠KBD + m∠DBC — adjacent angles
• m∠KBC = m∠ABK + 2·m∠ABK = 3·m∠ABK — substitute for m∠KBD and m∠DBC
• m∠AKB = m∠KBC — alternate interior angles of parallel lines AD, BC

• m∠AKB = 3·m∠ABK — substitute for m∠KBC

_____

Proof is always in the eye of the beholder, and the details depend on the supporting theorems and postulates you're allowed to invoke. The basic idea is that you have cut a vertex angle in half twice, and you're trying to show that the smallest part to the rest of it has the ratio 1 : 3.