Question

Pls help me solve for geometry

Answer 1

In rhombus PQRS, with PR as the diagonal dividing it into triangles PRS and PRQ, angles SPR and QPR are given. Opposite angles in a rhombus are equal, so
`\(\angle SPQ\)` is also
`\(63^\circ\).`

In the given rhombus PQRS, where PR is the diagonal dividing the rhombus into two triangles PRS and PRQ, you have angles SPR and QPR given as:

`\[ \angle SPR = 2x + 13 \]`

`\[ \angle QPR = 3x - 12 \]`

In a rhombus, opposite angles are equal. Therefore,
`\(\angle SPQ\)` is opposite to
`\(\angle SPR\)`, and
`\(\angle SPQ\)` is also opposite to
`\(\angle QPR\).`

`\[ \angle SPR = \angle SPQ \]`

`\[ \angle QPR = \angle SPQ \]`

Since
`\(\angle SPR\) and \(\angle QPR\)` are both opposite to
`\(\angle SPQ\)`, they must be equal.

2x + 13 = 3x - 12

Now, solve for x:

13 + 12 = 3x - 2x

25 = x

Now that we have the value of x, substitute it back into either of the angle expressions. Let's use
`\(\angle SPR\):`

`\[ \angle SPQ = 2x + 13 \]`

`\[ \angle SPQ = 2(25) + 13 \]`

`\[ \angle SPQ = 50 + 13 \]`

`\[ \angle SPQ = 63 \]`

Therefore,
`\( \angle SPQ = 63 \).`