Question

2/2 rectangles, rhombus, squares practice day

1. if abcd is a parallelogram, ( ad = 14 ), ( ec = 11 ), ( mangle abc = 64^circ ), ( mangle dac = 71^circ ), and ( mangle bdc = 25^circ ), find each measure.

a) ( bc = )

b) ( ac = )

c) ( mangle dab = )

d) ( mangle abd = )

e) ( mangle acd = )

f) ( mangle adb = )

Explanation:

Step1: Find BC (opposite sides ≅)

In parallelogram \(ABCD\), \(BC = AD = 14\)

Step2: Find AC (diagonals bisect each other)

Diagonals bisect, so \(AC = 2 \times EC = 2 \times 11 = 22\)

Step3: Find \(m\angle DAB\) (consecutive angles supplementary)

\(m\angle DAB + m\angle ABC = 180^\circ\)
\(m\angle DAB = 180^\circ - 64^\circ = 116^\circ\)

Step4: Find \(m\angle ABD\) (alternate interior angles)

\(AB \parallel DC\), so \(\angle ABD = \angle BDC = 25^\circ\)

Step5: Find \(m\angle ACD\) (alternate interior angles)

\(AD \parallel BC\), so \(\angle ACD = \angle DAC = 71^\circ\)

Step6: Find \(m\angle ADB\) (triangle angle sum)

First, \(\angle ADB = \angle DBC\). In \(\triangle ABC\), \(\angle BAC = 116^\circ - 71^\circ = 45^\circ\), so \(\angle ACB = 45^\circ\). Then \(m\angle DBC = 180^\circ - 64^\circ - 71^\circ = 45^\circ\)? No, correct: In \(\triangle ABD\), \(\angle ADB = 180^\circ - 116^\circ - 25^\circ = 39^\circ\)

Answer:

a) \(BC = 14\)
b) \(AC = 22\)
c) \(m\angle DAB = 116^\circ\)
d) \(m\angle ABD = 25^\circ\)
e) \(m\angle ACD = 71^\circ\)
f) \(m\angle ADB = 39^\circ\)