Question: The pentagon ABCDE is the frame for Ibrahim's mountain bike. ABC is an isosceles triangle in which...

Question

The pentagon ABCDE is the frame for Ibrahim's mountain bike.
ABC is an isosceles triangle in which AB=BC and angle BCA=65°.
In the quadrilateral, ACDE angle ACD=70°, angle CAE=90°, and AC is parallel to ED.
(a.) (i) Calculate the size of the angle ABC.
(ii) What facts about the angles of a triangle did you use in your calculation?
(b). Calculate the size of the angle CDE.

Answer 1

Solution

A pentagon is a 5-sided Polygon. A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). A convex pentagon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. When any internal angle is greater than 180° it is concave.

Complete step by step solution:
(a). Given:
$AB = BC$
∠BCA=∠BAC= $65^\circ$
So,∠BCA+∠BAC+∠ABC= $180∘$
$65^\circ + 65^\circ +$ ∠ABC= $180^\circ$
∠ABC= $180^\circ - 130^\circ$

∠ABC= $50$

(b). ∠CDE+∠ACD= $180^\circ$
(Where AC||ED)
∠CDE+ $70^\circ = 180^\circ$

∠CDE= $110^\circ$

Note: The properties of a simple pentagon (5-gon) are it must have five straight sides that meet to create five vertices but do not self-intersect: Pentagons have five straight sides. Pentagons have five interior angles, which sum to 540° The five sides do not intersect.

Additional information: There are two types of pentagon on the basis of sides and angles:
a. Regular pentagon
b. Irregular pentagon

Regular pentagon: A pentagon which has all five sides equal in length and eventually, all the five angles are equal in measure, is known as regular pentagon. Each interior angle of a regular pentagon is 1080.

Irregular pentagon: An irregular pentagon is a pentagon which does not have all sides equal. Also, it does not have all angles equal.

Pentagons can be regular or irregular and convex or concave. A regular pentagon is one with all equal sides and angles. Its interior angles are 108 degrees and its exterior angles are 72 degrees. An irregular pentagon is a shape that does not have equal sides and/or angles and therefore, does not have specified angles.

A convex pentagon is one whose vertices, or points where the sides meet, are pointing outwards as opposed to a concave pentagon whose vertices point inwards.