Question

1. the figure below shows a circle with center v and diameter zg. determine which of the following must be right - angles: ∠zap, ∠zav, ∠zac, ∠zcu, ∠pcu, ∠zpc.

1. the figure below shows a circle with center r and diameter ay, and tangent de. determine which of the following must be right - angles: ∠yde, ∠aty, ∠zadv, ∠ady, ∠zada, ∠rad, ∠yad, ∠atd.

1. the figure below shows a circle with center l and diameter hm, secant en, and tangent ft. determine which of the following must be right - angles: ∠eof, ∠len, ∠elf, ∠pfn, ∠lfo, ∠leo.

1. the figure below shows a circle with center z and diameter dl. determine which of the following must be right - angles: ∠zlo, ∠dol, ∠zlz, ∠ydl, ∠yde, ∠ydo.

Explanation:

Step1: Recall circle - angle properties

An angle inscribed in a semi - circle is a right angle. Also, a tangent to a circle is perpendicular to the radius at the point of tangency.

Step2: Analyze 1st figure

For the circle with center \(V\) and diameter \(ZG\), \(\angle ZPG\) is an inscribed angle subtended by the diameter \(ZG\). So \(\angle ZPG = 90^{\circ}\). Also, if \(PU\) is tangent to the circle at \(P\), then \(\angle ZPU=90^{\circ}\) and \(\angle GPU = 90^{\circ}\).

Step3: Analyze 2nd figure

For the circle with center \(R\) and diameter \(XY\), \(\angle XDY\) is an inscribed angle subtended by the diameter \(XY\), so \(\angle XDY = 90^{\circ}\). And if \(DE\) is tangent to the circle at \(D\), then \(\angle RDE=90^{\circ}\) and \(\angle ADY\) is not a right - angle (it is an inscribed angle that does not subtend a diameter), \(\angle ZAD\) is not a right - angle, \(\angle RAD\) is not a right - angle, \(\angle YAD\) is not a right - angle, \(\angle AYD\) is not a right - angle (it is an inscribed angle that does not subtend a diameter).

Step4: Analyze 3rd figure

For the circle with center \(L\), if \(FT\) is tangent to the circle at \(F\), then \(\angle LFT = 90^{\circ}\). \(\angle EOF\) is a central angle and not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle ENF\) is an inscribed angle and not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle ELF\) is a central angle and not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle FPN\) is not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle LFO\) is not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle LEO\) is not a right - angle (no information to suggest it is \(90^{\circ}\)).

Step5: Analyze 4th figure

For the circle with center \(Z\) and diameter \(DL\), \(\angle DOL\) is a central angle and not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle ZOL\) is not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle ZDL\) is an inscribed angle subtended by the diameter \(DL\), so \(\angle ZDL=90^{\circ}\), \(\angle YDE\) is not a right - angle (no information to suggest it is \(90^{\circ}\)), \(\angle YDO\) is not a right - angle (no information to suggest it is \(90^{\circ}\)).

1. Right - angles: \(\angle ZPG,\angle ZPU,\angle GPU\)
2. Right - angles: \(\angle XDY,\angle RDE\)
3. Right - angle: \(\angle LFT\)
4. Right - angle: \(\angle ZDL\)

Answer:

1. \(\angle ZPG,\angle ZPU,\angle GPU\)
2. \(\angle XDY,\angle RDE\)
3. \(\angle LFT\)
4. \(\angle ZDL\)