Figure It Out: Squares

Angles in a Square (Page 93)

Question: What are the measures of $\angle 1, \angle 2, \angle 3, \angle 4$ formed by the diagonal?

Solution: In a square, the angle at the vertex is $90^\circ$. The sides are equal ($AD = DC$), so the triangle formed by the diagonal ($\Delta ADC$) is an isosceles right-angled triangle.

• $\angle 1 = \angle 3$.
• $\angle 1 + \angle 3 + 90^\circ = 180^\circ$.
• $2(\angle 1) = 90^\circ \Rightarrow \angle 1 = 45^\circ$.
• Answer: All such angles ($\angle 1, \angle 2, \angle 3, \angle 4$) are $45^\circ$.

Figure It Out (Page 94)

Question 1: Find all other angles inside the rectangles.

• (i) Diagonals intersect at $30^\circ$ (vertical).
  • Side angle = $(180 - 30)/2 = 75^\circ$.
  • Complementary angle = $90 - 75 = 15^\circ$.
• (ii) Diagonals intersect at $110^\circ$.
  • Vertical opposite = $110^\circ$. Adjacent linear pair = $70^\circ$.
  • Base angles for $110^\circ$ triangle = $(180-110)/2 = 35^\circ$.
  • Base angles for $70^\circ$ triangle = $(180-70)/2 = 55^\circ$.

Question 3: Circle Diameters

• Problem: $PL$ and $AM$ are perpendicular diameters. What is $APML$?
• Reasoning:
  • Diameters are equal ($PL = AM$).
  • They bisect each other (at center $O$).
  • They are perpendicular ($90^\circ$).
• Conclusion: A quadrilateral with equal, bisecting, perpendicular diagonals is a Square.