ABCD is a trapezium with AB || BC, AB=AD and C = 180°. Determine the unknown angles.

Here's the step-by-step solution for problem (1):

Problem (1): Trapezium ABCD with given properties

Step 1: Interpret the given information and correct typos.

• ABCD is a trapezium.
• The problem states $AB \parallel BC$, which is a typo. For a trapezium, one pair of opposite sides must be parallel. From the diagram, it's clear that $AB \parallel DC$. We will proceed with this assumption.
• $AB = AD$. This means $\triangle ABD$ is an isosceles triangle.
• $\angle C = 80^\circ$. (The text says $180^\circ$, which is impossible for an angle in a quadrilateral; we use the diagram's $80^\circ$).
• We need to find the unknown angles: $a = \angle CDA$, $b = \angle ABD$, $d = \angle ABC$, $e = \angle DAB$.

Step 2: Use the property of parallel lines ($AB \parallel DC$). Since $AB \parallel DC$, the consecutive interior angles on the transversal $BC$ are supplementary: $\angle ABC + \angle BCD = 180^\circ$ $\angle d + 80^\circ = 180^\circ$ $\angle d = 180^\circ - 80^\circ$ $\angle d = \mathbf{100^\circ}$

Also, since $AB \parallel DC$, the alternate interior angles formed by the transversal $BD$ are equal: $\angle ABD = \angle BDC$ $\angle b = \angle BDC$

Step 3: Use the property of the isosceles triangle $\triangle ABD$. Given $AB = AD$, $\triangle ABD$ is an isosceles triangle. The base angles opposite the equal sides are equal: $\angle ABD = \angle ADB$ $\angle b = \angle ADB$

Step 4: Determine angle $a$ ($\angle CDA$). From Step 2 and Step 3, we have $\angle ADB = b$ and $\angle BDC = b$. Therefore, $\angle CDA$ is the sum of these two angles: $\angle CDA = \angle ADB + \angle BDC$ $\angle a = b + b$ $\angle a = 2b$

Step 5: Determine angle $e$ ($\angle DAB$). In $\triangle ABD$, the sum of angles is $180^\circ$: $\angle DAB + \angle ABD + \angle ADB = 180^\circ$ $\angle e + b + b = 180^\circ$ $\angle e + 2b = 180^\circ$

Step 6: Use the property of an isosceles trapezium. In a trapezium where the non-parallel sides are equal, it is an isosceles trapezium. In this case, the condition $AB=AD$ along with $AB \parallel DC$ and $\angle C = 80^\circ$ implies that $AD=BC$, making it an isosceles trapezium. In an isosceles trapezium, the base angles are equal: $\angle CDA = \angle BCD$ $\angle a = 80^\circ$

Step 7: Solve for $b$ and $e$ using the value of $a$. From Step 4, we have $\angle a = 2b$. Substitute $\angle a = 80^\circ$: $80^\circ = 2b$ $b = \frac{80^\circ}{2}$ $b = \mathbf{40^\circ}$

From Step 5, we have $\angle e + 2b = 180^\circ$. Substitute $2b = 80^\circ$: $\angle e + 80^\circ = 180^\circ$ $\angle e = 180^\circ - 80^\circ$ $\angle e = \mathbf{100^\circ}$

Summary of unknown angles:

• $\angle a = \angle CDA = \mathbf{80^\circ}$
• $\angle b = \angle ABD = \mathbf{40^\circ}$
• $\angle d = \angle ABC = \mathbf{100^\circ}$
• $\angle e = \angle DAB = \mathbf{100^\circ}$