Subjects geometry

Angle Quad

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Angle Quad


1. **Problem Statement:** Find the angles $\angle LADE$, $\angle LABE$, and $\angle LBED$ in the given quadrilateral with vertices $O$, $D$, $B$, $C$, and point $E$ on segment $OC$. Given:\n- $\angle D = 133^\circ$\n- $\angle C = 22^\circ$\n- $OB \parallel DC$\n\n2. **Step 1: Analyze Given Information**\nSince $OB \parallel DC$, corresponding and alternate interior angles involving these lines are equal.\n\n3. **Step 2: Use Angle Sum Property of Quadrilateral**\nThe sum of interior angles in any quadrilateral is $360^\circ$. Therefore,\n$$ \angle O + \angle D + \angle B + \angle C = 360^\circ $$\nGiven $\angle D = 133^\circ$ and $\angle C = 22^\circ$, so\n$$ \angle O + \angle B + 155^\circ = 360^\circ $$\n$$ \angle O + \angle B = 205^\circ $$\n\n4. **Step 3: Use Parallel Line Angles**\nSince $OB \parallel DC$, $\angle O = \angle D = 133^\circ$ (alternate interior angles), so\n$$ \angle B = 205^\circ - 133^\circ = 72^\circ $$\n\n5. **Step 4: Identify $\angle LADE$**\nPoint $A$ lies on lines from $L$, $D$, and $E$. $\angle LADE$ is the angle at $D$ formed by points $L$ and $E$. Since $E$ is on $OC$ and $OB \parallel DC$, the corresponding alternate interior angle at $D$ is equal to $\angle LABE$, which we identify next.\n\n6. **Step 5: Find $\angle LABE$ and $\angle LBED$**\nSince $E$ lies on $OC$, and $OB \parallel DC$, angles at $B$ and $D$ correspond. We can conclude:\n- $\angle LADE = 22^\circ$ (corresponding to $\angle C$)\n- $\angle LABE = 133^\circ$ (corresponding to $\angle D$)\n- $\angle LBED = 72^\circ$ (from step 4 calculation)\n\n**Final Answers:**\n$$ \angle LADE = 22^\circ $$\n$$ \angle LABE = 133^\circ $$\n$$ \angle LBED = 72^\circ $$