Question

find the measures of the indicated angles below.
given ab || cd, t = 56° image not drawn to scale
m∠dca=

m∠acb=

m∠cab=

m∠abc=

Explanation:

Step1: Find \(m\angle DCA\)

Since \(AB\parallel CD\), \(\angle DCA=\angle1 = 56^{\circ}\) (alternate - interior angles).

Step2: Find \(m\angle CAB\)

Assume the figure is a rectangle (right - angled at \(D\)). In right - triangle \(ADC\), \(\angle DAC = 90^{\circ}-\angle DCA\). But if we consider the property of the rectangle (opposite sides are parallel and all angles are \(90^{\circ}\)), and using the fact that \(\angle1 = 56^{\circ}\), in right - triangle \(ABC\) (if we assume \(AD\perp DC\) and \(AB\parallel DC\)), \(\angle CAB=\angle1 = 56^{\circ}\) (alternate - interior angles for \(AB\parallel CD\) and transversal \(AC\)).

Step3: Find \(m\angle ACB\)

In right - triangle \(ABC\) (assuming \(AB\perp BC\)), \(\angle ABC = 90^{\circ}\). Using the angle - sum property of a triangle (\(\angle CAB+\angle ABC+\angle ACB=180^{\circ}\)). Substitute \(\angle CAB = 56^{\circ}\) and \(\angle ABC=90^{\circ}\). Then \(56^{\circ}+90^{\circ}+\angle ACB = 180^{\circ}\). So \(\angle ACB=180^{\circ}-(56^{\circ} + 90^{\circ})=34^{\circ}\).

Step4: Find \(m\angle ABC\)

Assuming \(AB\perp BC\) (if the figure is a rectangle - like structure with \(AB\parallel CD\) and \(AD\parallel BC\) and right - angles at \(B\) and \(D\)), \(m\angle ABC = 90^{\circ}\).

Answer:

\(m\angle ABC = 90^{\circ}\), \(m\angle CAB=56^{\circ}\), \(m\angle ACB = 34^{\circ}\), \(m\angle DCA=56^{\circ}\)