Prove that the diagonal of a rhombus divides the rhombus into two congruent triangles. Also,
prove that the diagonal of rhombus bisects its vertex angles.
Answers
Step-by-step explanation:
First, all four sides of a rhombus are congruent, meaning that if we find one side, we can simply multiply by four to find the perimeter. Second, the diagonals of a rhombus are perpendicular bisectors of each other, thus giving us four right triangles and splitting each diagonal in half.
Answer:
A rhombus has four sides and all are equal.
i.e. AB = BC = CD = AD
Now, take diagonal AC. This divides the rhombus into two triangles ABC and CDA.
In ΔABC and ΔCDA,
AB = CD {equal sides}
BC = DA {equal sides}
AC = AC {common}
By SSS congruent criterian,
ΔABC ≅ ΔCDA
Now, ΔABC and ΔCDA are isoscales triangle since they have two equal sides
Thus, ∠DAC = ∠DCA = ∠BAC = ∠BCA = DAB/2
Again, take diagonal BD, we can show that,
ΔABD ≅ ΔCBD
and ∠ADB = ∠CDB = ∠ABD = ∠CBD = ABC/2
Let E is the mid point where diagonal AC and BD intersect each other.
So, ∠DAE = ∠BAE = ∠BCE = ∠DCE = ABC/2
and ∠ABE = ∠CBE = ∠CDE = ∠ADE = ABD/2
and AB = BC = CD = AD
Thus, the four triangle are congrent by AAS.
hence, the diagonals of a rhombus divide it into 4 equal congruent triangles.