Question

1. abcd is a rectangle

m∠1=
m∠2=
m∠3=
m∠4=
m∠5=
m∠6=
m∠7=

Explanation:

Step1: Find \(m\angle2\)

Vertical angles are equal. So \(m\angle2 = 116^{\circ}\)

Step2: Find \(m\angle1\)

In \(\triangle BCD\), \(\angle1+\angle4+\angle BCD = 180^{\circ}\). Also, in a rectangle, diagonals are equal and bisect each other. \(\angle BCD = 90^{\circ}\). \(\angle1=\angle4\) (isosceles triangle as \(BC = CD\) in rectangle properties). \(m\angle1=\frac{180 - 116}{2}=32^{\circ}\)

Step3: Find \(m\angle3\)

\(\angle3 = 90 - 32=58^{\circ}\) (complementary angles in right - angled triangle \(ABC\))

Step4: Find \(m\angle4\)

\(m\angle4 = 32^{\circ}\) (from step 2)

Step5: Find \(m\angle5\)

\(\angle5=\angle3 = 58^{\circ}\) (alternate interior angles for parallel sides \(AD\parallel BC\) and transversal \(AC\))

Step6: Find \(m\angle6\)

\(\angle6=\angle3 = 58^{\circ}\) (alternate interior angles for parallel sides \(AB\parallel CD\) and transversal \(AC\))

Step7: Find \(m\angle7\)

\(\angle7=\angle1 = 32^{\circ}\) (alternate interior angles for parallel sides \(AB\parallel CD\) and transversal \(BD\))

Answer:

\(m\angle1 = 32^{\circ}\), \(m\angle2 = 116^{\circ}\), \(m\angle3 = 58^{\circ}\), \(m\angle4 = 32^{\circ}\), \(m\angle5 = 58^{\circ}\), \(m\angle6 = 58^{\circ}\), \(m\angle7 = 32^{\circ}\)