المثال 2 - كتاب الرياضيات - الصف 10 - الفصل 2 - المملكة العربية السعودية

المثال 2

جبر: أوجد كل قياس مما يأتي:

m∠R

m∠S

m∠A

m∠C

المثال 3

برهان: اكتب برهانًا ذا عمودين. المعطيات: m∠T = ½ m∠S المطلوب: mTUR = 2m US

المثال 4

جبر: أوجد قيمة كل مما يأتي:

x

x

m∠T

m∠C

المثال 5

جبر: أوجد كل قياس مما يأتي:

m∠T

m∠H

m∠Z

m∠G

برهان: اكتب برهانًا حرًا للنظرية 8.9.

برهان: برهن النظرية 8.6 لحالتي الزاوية المحيطية في الدائرة فيما يأتي:

الحالة الثانية: المعطيات: يقع المركز P داخل ∠ABC BD قطر للدائرة. المطلوب: m∠ABC = ½ mAC

الحالة الثالثة: المعطيات: يقع المركز P خارج ∠ABC BD قطر للدائرة. المطلوب: m∠ABC = ½ mAC

برهان: اكتب برهانًا من النوع المحدد لكل من النظريتين الآتيتين:

النظرية 8.7، برهانًا ذا عمودين.

النظرية 8.8، برهانًا حرًا.

وزارة التعليم Ministry of Education 2025 - 1447

206 الفصل 8 الدائرة

--- SECTION: المثال 2 --- المثال 2 جبر: أوجد كل قياس مما يأتي: --- SECTION: 14 --- m∠R --- SECTION: 15 --- m∠S --- SECTION: 16 --- m∠A --- SECTION: 17 --- m∠C --- SECTION: المثال 3 --- المثال 3 --- SECTION: 18 --- برهان: اكتب برهانًا ذا عمودين. المعطيات: m∠T = ½ m∠S المطلوب: mTUR = 2m US --- SECTION: المثال 4 --- المثال 4 جبر: أوجد قيمة كل مما يأتي: --- SECTION: 19 --- x --- SECTION: 20 --- x --- SECTION: 21 --- m∠T --- SECTION: 22 --- m∠C --- SECTION: المثال 5 --- المثال 5 جبر: أوجد كل قياس مما يأتي: --- SECTION: 23 --- m∠T --- SECTION: 24 --- m∠H --- SECTION: 25 --- m∠Z --- SECTION: 26 --- m∠G --- SECTION: 27 --- برهان: اكتب برهانًا حرًا للنظرية 8.9. برهان: برهن النظرية 8.6 لحالتي الزاوية المحيطية في الدائرة فيما يأتي: --- SECTION: 28 --- الحالة الثانية: المعطيات: يقع المركز P داخل ∠ABC BD قطر للدائرة. المطلوب: m∠ABC = ½ mAC --- SECTION: 29 --- الحالة الثالثة: المعطيات: يقع المركز P خارج ∠ABC BD قطر للدائرة. المطلوب: m∠ABC = ½ mAC برهان: اكتب برهانًا من النوع المحدد لكل من النظريتين الآتيتين: --- SECTION: 30 --- النظرية 8.7، برهانًا ذا عمودين. --- SECTION: 31 --- النظرية 8.8، برهانًا حرًا. وزارة التعليم Ministry of Education 2025 - 1447 206 الفصل 8 الدائرة --- VISUAL CONTEXT --- **DIAGRAM**: Untitled Description: A circle containing an inscribed quadrilateral ABCD. Chords AC and BD intersect inside the circle. The measures of angles and arcs are given as algebraic expressions. Specifically, m∠B = (7x-8)°, m∠C = (5y-3)°, m∠D = (4y+7)°, and the arc AD = 5x°. Data: Geometric relationships between inscribed angles and intercepted arcs. Context: Used to find angle measures in an inscribed quadrilateral. **DIAGRAM**: Untitled Description: A circle with chords QS and RT intersecting inside the circle. The measure of an inscribed angle is 32°. The intercepted arc RS is (5x+4)° and the intercepted arc QT is (6x-2)°. Data: Geometric relationships between intersecting chords and intercepted arcs. Context: Used to find angle measures related to intersecting chords. **DIAGRAM**: Untitled Description: A circle with an inscribed quadrilateral RSTU. A center point is indicated. This diagram is for a proof problem. Data: Geometric configuration for a proof involving angles and arcs in an inscribed quadrilateral. Context: Used as a visual aid for a two-column proof. **DIAGRAM**: Untitled Description: A circle with an inscribed triangle RST. The side RT is a diameter of the circle. The angle at R is x° and the angle at T is 2x°. A center point is indicated on the diameter RT. Data: Geometric relationships in a triangle inscribed in a semicircle. Context: Used to find the value of x and angle measures in an inscribed triangle. **DIAGRAM**: Untitled Description: A circle with an inscribed triangle CDE. A center point is indicated. The angle at E is 3x° and the intercepted arc CD is (5x-12)°. Data: Geometric relationships between an inscribed angle and its intercepted arc. Context: Used to find the value of x and angle measures. **DIAGRAM**: Untitled Description: A circle with an inscribed quadrilateral RWTZ. A center point is indicated. The angle at W is 45°, the angle at R is 4x°, and the angle at Z is (2x+30)°. Data: Geometric relationships in an inscribed quadrilateral. Context: Used to find angle measures in an inscribed quadrilateral. **DIAGRAM**: Untitled Description: A circle with an inscribed quadrilateral GHJK. A center point is indicated. The angle at J is (3y+9)°, the angle at K is (x+21)°, the arc HG is 2x°, and the arc GK is (4y-11)°. Data: Geometric relationships in an inscribed quadrilateral involving angles and arcs. Context: Used to find angle and arc measures in an inscribed quadrilateral. **DIAGRAM**: Untitled Description: A circle with points A, B, C, D. Point P is the center of the circle. BD is a diameter. The angle ∠ABC is inscribed in the circle, and the center P lies inside ∠ABC. This diagram illustrates Case Two for proving Theorem 8.6. Data: Visual representation for a geometric proof. Context: Used as a visual aid for proving Theorem 8.6, Case Two, regarding inscribed angles. **DIAGRAM**: Untitled Description: A circle with points A, B, C, D. Point P is the center of the circle. BD is a diameter. The angle ∠ABC is inscribed in the circle, and the center P lies outside ∠ABC. This diagram illustrates Case Three for proving Theorem 8.6. Data: Visual representation for a geometric proof. Context: Used as a visual aid for proving Theorem 8.6, Case Three, regarding inscribed angles.