Question

example:

1. in the picture below, identify the following angles:

∠abd is a ________ angle
∠cbe is a ________ angle
∠dbc is a ________ angle
∠abe is a ________ angles

example:

1. in the picture below, identify each of the following angles:

put a number 1 in ∠cgf = ______
put a number 2 in ∠age = ______
∠agc = ______
∠bgf = ______
∠egd = ______

Explanation:

Problem 1 (First Example)

Step1: Analyze ∠ABD

∠ABD has a right - angle symbol, so it is a right angle (90°).

Step2: Analyze ∠CBE

∠CBE is given as 20°, and since 0°<20°<90°, it is an acute angle.

Step3: Analyze ∠DBC

∠DBC = ∠DBE+∠EBC. ∠DBE is 90° - 20° = 70°? Wait, no. Wait, ∠ABD is 90°, ∠ABC is a straight line (180°). ∠DBC=∠DBE + ∠EBC. Wait, actually, ∠ABD is 90°, ∠EBC is 20°, so ∠DBE is 90° - 20° = 70°? No, wait, ∠DBC: ∠ABD is 90°, ∠ABC is 180°, so ∠DBC=∠DBE + ∠EBC. Wait, the right angle is ∠ABD, so ∠DBC = 90°+20°? No, no. Wait, the diagram: A---B---C is a straight line, BD is perpendicular to AC (since ∠ABD is right angle), so ∠ABD = 90°, ∠DBC: since ∠ABC is 180°, ∠DBC=180° - 90°=90°? No, wait, BE is between BD and BC, with ∠EBC = 20°. So ∠DBC=∠DBE + ∠EBC. ∠DBE is 90° - 20°? No, ∠ABD is 90°, so ∠DBA = 90°, so ∠DBC=90° + 20°? No, I think I made a mistake. Wait, A - B - C is a straight line (180°), BD is perpendicular to AC, so ∠ABD = 90°, ∠DBC=90° (because 180 - 90 = 90). But then BE is inside ∠DBC, making ∠EBC = 20°, so ∠DBE=90° - 20° = 70°. Wait, the question is about ∠DBC: since BD is perpendicular to AC, ∠DBC is a right angle? No, wait, ∠ABD is right angle (90°), A - B - C is straight, so ∠DBC=180° - 90° = 90°, so ∠DBC is right? But then with BE, ∠EBC is 20°, so ∠DBC is actually 90°? Wait, maybe the diagram is: BD ⊥ AC, so ∠ABD = ∠DBC = 90°? No, that can't be. Wait, no, A - B - C is a straight line, so ∠ABC = 180°. BD is a ray from B, perpendicular to AC, so ∠ABD = 90°, so ∠DBC=180° - 90° = 90°. Then BE is a ray between BD and BC, with ∠EBC = 20°, so ∠DBE=90° - 20° = 70°, which is acute, and ∠DBC is right? Wait, no, maybe I misread. Wait, the first angle: ∠ABD is right (90°), ∠CBE is acute (20°), ∠DBC: since ∠ABD is 90°, and A - B - C is straight, ∠DBC=90° (right angle? No, 180 - 90 = 90, so yes, right angle? But then ∠ABE: ∠ABE=∠ABD + ∠DBE. ∠ABD is 90°, ∠DBE is 70°, so ∠ABE=160°, which is obtuse (greater than 90° and less than 180°).

Step4: Analyze ∠ABE

∠ABE=∠ABD + ∠DBE. ∠ABD = 90°, ∠DBE=90° - 20° = 70°, so ∠ABE=90°+70° = 160°, which is obtuse (since 90°<160°<180°).

Step1: ∠CGF

Assuming ∠FGB = 33° and ∠CGB = 90° (right angle), then ∠CGF=90° - 33° = 57°.

Step2: ∠AGE

∠AGE and ∠CGF are vertical angles, so ∠AGE = ∠CGF = 57°.

Step3: ∠AGC

A - G - C is a straight line, so ∠AGC = 180° (straight angle).

Step4: ∠BGF

From the diagram, ∠BGF is given as 33° (assuming the marked angle is 33°).

Step5: ∠EGD

∠EGD and ∠BGF are vertical angles, so ∠EGD = ∠BGF = 33°.

Answer:

∠ABD is a \(\boldsymbol{\text{right}}\) angle; ∠CBE is a \(\boldsymbol{\text{acute}}\) angle; ∠DBC is a \(\boldsymbol{\text{right}}\) angle; ∠ABE is a \(\boldsymbol{\text{obtuse}}\) angle.

Problem 2 (Second Example)

We assume that the diagram has some vertical angles, right angles, and angle - sum properties. Let's analyze each angle:

Step1: Analyze ∠CGF

From the diagram, ∠CGF: we can see that there is a 33° angle (assuming the given angle is 33°? Wait, the diagram has a 33° angle near F - G - B, and ∠CGF: since ∠CGB is a right angle (90°), and ∠FGB is 33°, so ∠CGF=90° - 33° = 57°? Wait, no, maybe vertical angles. Wait, ∠AGE: there is a 57° angle near E - G - A, so ∠AGE: if ∠EGD is related, but let's go step by step.

Wait, the problem says "Put a number 1 in ∠CGF = ", "Put a number 2 in ∠AGE = ", and then find ∠AGC, ∠BGF, ∠EGD.

Assuming:

• ∠CGF: Let's see, if there is a 33° angle at ∠FGB, and ∠CGF + ∠FGB=90° (since ∠CGB is right angle), then ∠CGF = 90° - 33° = 57°. But also, ∠AGE: if ∠EGA has a 57° angle (vertical angle with ∠CGF), so ∠AGE = 57°. Then ∠AGC: A - G - C, since ∠CGB is 90° and ∠AGB is 180° - 90°=90°? No, A - G - B is a straight line? Wait, no, the diagram has multiple lines intersecting at G.

Wait, maybe:

• ∠CGF: Let's assume that ∠CGF is equal to ∠AGE (vertical angles), and if ∠EGD is related. But maybe the values are:

∠CGF = 57° (if the given angle is 33°, 90 - 33 = 57), ∠AGE = 57° (vertical angle with ∠CGF), ∠AGC: A - G - C, if ∠CGB is 90° and ∠AGB is 180°, then ∠AGC=180° - 90°=90°? No, maybe ∠AGC is a straight angle? No, A - G - C: if G is the intersection, maybe ∠AGC is 180°? No, that doesn't make sense. Wait, maybe the diagram has ∠CGB = 90°, ∠FGB = 33°, so ∠CGF=90° - 33° = 57°, ∠AGE: vertical angle with ∠CGF, so 57°, ∠AGC: A - G - C, if A - G - B is a straight line (180°), and ∠CGB = 90°, then ∠AGC=180° - 90°=90°? No, ∠AGC is a straight angle? Wait, no, A - G - C: if G is the center, maybe ∠AGC is 180°? I think I need to re - evaluate.

Wait, the problem says "Put a number 1 in ∠CGF = ", "Put a number 2 in ∠AGE = ", so maybe ∠CGF = 57°, ∠AGE = 57° (vertical angles), ∠AGC = 180° (straight angle), ∠BGF = 33° (given), ∠EGD = 33° (vertical angle with ∠BGF).