Question

$\bf \huge {\underline {\underline \red{QuEsTiOn}}}$

In a rhombus ABCD show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.​

Answer 1

Answer:

Inrhombusallsideareequalanddiagonolsaredifferentbutintersectat90

⇒AB=BC=CD=DA

⇒AC⊥BD

InΔABC

⇒AB=BC

⇒∴∠CAB=∠ACB

⇒AO=OC

∴∠ABO=∠CBO

andsimillarlyinΔADC

⇒AD=DC

⇒AO=OC

∴correspondingangleareequal

⇒∠DAC=∠DCA

and∠CDO=∠ADO

Hence,AC bisect ∠Aand∠C and BD bisects ∠Band∠D

Step-by-step explanation:

Hope this helps

Answer 2

Question :-

In a rhombus ABCD show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D.

To Prove :-

diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D

Construction :-

• Join A and C
• Join B and D

Proof :-

In rhombus all sides are equal

so, AB = BC = CD = DA

In △ABC and △ADC

AB = DA

BC = CD

AC = AC

△ABC ≅ △ADC [ By SSS congruence rule]

∠BAC = ∠DAC [ By CPCT ]

∠BCA = ∠DCA [ By CPCT ]

So, AC bisects ∠A as well as ∠C.

In △DAB and △DCB

DA = CD

AB = BC

BD = BD

△DAB ≅ △DCB [ By SSS congruence rule]

∠ADB = ∠CDB [ By CPCT ]

∠ABD = ∠CBD [ By CPCT ]

So, BD bisects ∠B as well as ∠D.

Additionally✏

• SSS congruence rule - If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

• SAS congruence rule - If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

• ASA congruence rule - If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

• AAS congruence rule - If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

• RHS congruence rule - If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

• CPCT stands for corresponding parts of congruent triangle.