Finding the Value of Angle $x$ in a Concave Quadrilateral

Maths Question
1

Don t Use Pen and raper
50°
110°
C=?

Soru Görseli
Soru Görseli800×851 61.4 KB

2

Finding the Value of Angle x in a Concave Quadrilateral

:light_bulb: [RULE / FORMULA USED:] The Exterior Angle Property of a Concave Quadrilateral (Arrowhead Property) states that the interior reflex angle (or the interior non-reflex angle pointing outward) is equal to the sum of the three interior opposite angles.
Formula: \text{Angle } C = \text{Angle } A + \text{Angle } B + \text{Angle } D

:brain: [SOLUTION STEPS:]

Step 1 — Identify the Given Values
In the provided image, we have a concave quadrilateral ABCD. The internal angles given are:

  • Angle B = 50^\circ
  • Angle D = 50^\circ
  • The interior angle at C = 110^\circ
  • The angle at A = x

Step 2 — Apply the Arrowhead Theorem
According to the geometric property of this shape, the angle located at the “indent” (Angle C) is the sum of the three pointed angles (A, B, and D).
Equation: 110^\circ = x + 50^\circ + 50^\circ

Step 3 — Simplify and Solve for x
First, add the known angles on the right side:
50^\circ + 50^\circ = 100^\circ
Now, the equation is:
110^\circ = x + 100^\circ

Subtract 100^\circ from both sides to isolate x:
x = 110^\circ - 100^\circ
x = 10^\circ

━━━━━━━━━━━━━━━━━━━━━━━━━━━
:white_check_mark: [ANSWER:] x = 10^\circ
━━━━━━━━━━━━━━━━━━━━━━━━━━━

:bullseye: [KEY CONCEPTS:]

1. Concave Quadrilateral (Arrowhead)

  • [Definition:] A quadrilateral where one interior angle is greater than 180^\circ (reflex angle).
  • [In this problem:] The shape ABCD forms an arrowhead where the exterior-facing angle at C is the sum of the other three.

2. Sum of Interior Angles

  • [Definition:] The sum of interior angles of any quadrilateral is always 360^\circ.
  • [In this problem:] If you consider the reflex angle at C (360^\circ - 110^\circ = 250^\circ), then x + 50^\circ + 50^\circ + 250^\circ = 360^\circ, which also results in x = 10^\circ.

:warning: [COMMON MISTAKES:]

:cross_mark: Incorrect Shape Identification

  • [Wrong:] Treating the shape as a standard triangle and assuming x + 50 + 50 = 180.
  • [Right:] Recognizing this is a four-sided concave polygon, not a simple triangle.
  • [Why it’s wrong:] There are four vertices (A, B, C, D), making it a quadrilateral governed by different angle properties.

Would you like me to explain how to prove the Arrowhead Theorem using auxiliary lines?